[docs] add tutorial: Performance problems with sum-if formulations

docs/make.jl

@@ -317,6 +317,7 @@ const _PAGES = [
             "tutorials/getting_started/debugging.md",
             "tutorials/getting_started/design_patterns_for_larger_models.md",
             "tutorials/getting_started/performance_tips.md",
+            "tutorials/getting_started/sum_if.md",
         ],
         "Transitioning" =>
             ["tutorials/transitioning/transitioning_from_matlab.md"],

docs/src/tutorials/getting_started/sum_if.jl

@@ -0,0 +1,250 @@
+# Copyright (c) 2024 Oscar Dowson and contributors                               #src
+#                                                                                #src
+# Permission is hereby granted, free of charge, to any person obtaining a copy   #src
+# of this software and associated documentation files (the "Software"), to deal  #src
+# in the Software without restriction, including without limitation the rights   #src
+# to use, copy, modify, merge, publish, distribute, sublicense, and/or sell      #src
+# copies of the Software, and to permit persons to whom the Software is          #src
+# furnished to do so, subject to the following conditions:                       #src
+#                                                                                #src
+# The above copyright notice and this permission notice shall be included in all #src
+# copies or substantial portions of the Software.                                #src
+#                                                                                #src
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR     #src
+# IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,       #src
+# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE    #src
+# AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER         #src
+# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,  #src
+# OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE  #src
+# SOFTWARE.                                                                      #src
+
+# # Performance problems with sum-if formulations
+
+# The purpose of this tutorial is to explain a common performance issue that can
+# arise with summations like `sum(x[a] for a in list if condition(a))`. This
+# issue is particularly common in models with graph or network structures.
+
+# !!! tip
+#     This tutorial is more advanced than the other "Getting started" tutorials.
+#     It's in the "Getting started" section because it is one of the most common
+#     causes of performance problems that users experience when they first start
+#     using JuMP to write large scale programs. If you are new to JuMP, you may
+#     want to briefly skim the tutorial and come back to it once you have
+#     written a few JuMP models.
+
+# ## Required packages
+
+# This tutorial uses the following packages
+
+using JuMP
+import Plots
+
+# ## Data
+
+# As a motivating example, we consider a network flow problem, like the examples
+# in [Network flow problems](@ref) or [The network multi-commodity flow problem](@ref).
+
+# Here is a function that builds a random graph. The specifics do not matter.
+
+function build_random_graph(num_nodes::Int, num_edges::Int)
+    nodes = 1:num_nodes
+    edges = Pair{Int,Int}[i - 1 => i for i in 2:num_nodes]
+    while length(edges) < num_edges
+        edge = rand(nodes) => rand(nodes)
+        if !(edge in edges)
+            push!(edges, edge)
+        end
+    end
+    function demand(n)
+        if n == 1
+            return -1
+        elseif n == num_nodes
+            return 1
+        else
+            return 0
+        end
+    end
+    return nodes, edges, demand
+end
+
+nodes, edges, demand = build_random_graph(4, 8)
+
+# The goal is to decide the flow of a commodity along each edge in `edges` to
+# satisfy the `demand(n)` of each node `n` in `nodes`.
+
+# The mathematical formulation is:
+#
+# ```math
+# \begin{aligned}
+# s.t. && \sum_{(i,n)\in E} x_{i,n} - \sum_{(n,j)\in E} x_{n,j} = d_n && \forall n \in N\\
+# && x_{e} \ge 0 && \forall e \in E
+# \end{aligned}
+# ```
+
+# ## Naïve model
+
+# The first model you might write down is:
+
+model = Model()
+@variable(model, flows[e in edges] >= 0)
+@constraint(
+    model,
+    [n in nodes],
+    sum(flows[(i, j)] for (i, j) in edges if j == n) -
+    sum(flows[(i, j)] for (i, j) in edges if i == n) == demand(n)
+);
+
+# The benefit of this formulation is that it looks very similar to the
+# mathematical formulation of a network flow problem.
+
+# The downside to this formulation is subtle. Behind the scenes, the JuMP
+# `@constraint` macro expands to something like:
+
+model = Model()
+@variable(model, flows[e in edges] >= 0)
+for n in nodes
+    flow_in = AffExpr(0.0)
+    for (i, j) in edges
+        if j == n
+            add_to_expression!(flow_in, flows[(i, j)])
+        end
+    end
+    flow_out = AffExpr(0.0)
+    for (i, j) in edges
+        if i == n
+            add_to_expression!(flow_out, flows[(i, j)])
+        end
+    end
+    @constraint(model, flow_in - flow_out == demand(n))
+end
+
+# This formulation includes two for-loops, with a loop over every edge (twice) for
+# every node. The [big-O notation](https://en.wikipedia.org/wiki/Big_O_notation)
+# of the runtime is ``O(|nodes| \times |edges|)``. If
+# you have a large number of nodes and a large number of edges, the runtime of
+# this loop can be large.
+
+# Let's build a function to benchmark our formulation:
+
+function build_naive_model(nodes, edges, demand)
+    model = Model()
+    @variable(model, flows[e in edges] >= 0)
+    @constraint(
+        model,
+        [n in nodes],
+        sum(flows[(i, j)] for (i, j) in edges if j == n) -
+        sum(flows[(i, j)] for (i, j) in edges if i == n) == demand(n)
+    )
+    return model
+end
+
+nodes, edges, demand = build_random_graph(1_000, 2_000)
+@elapsed build_naive_model(nodes, edges, demand)
+
+# A good way to benchmark is to measure the runtime across a wide range of input
+# sizes. From our big-O analysis, we should expect that doubling the number of
+# nodes and edges results in a 4x increase in the runtime.
+
+run_times = Float64[]
+factors = 1:10
+for factor in factors
+    GC.gc()  #src
+    graph = build_random_graph(1_000 * factor, 5_000 * factor)
+    push!(run_times, @elapsed build_naive_model(graph...))
+end
+Plots.plot(; xlabel = "Factor", ylabel = "Runtime [s]")
+Plots.scatter!(factors, run_times; label = "Actual")
+a, b = hcat(ones(10), factors .^ 2) \ run_times
+Plots.plot!(factors, a .+ b * factors .^ 2; label = "Quadratic fit")
+
+# As expected, the runtimes demonstrate quadratic scaling: if we double the
+# number of nodes and edges, the runtime increases by a factor of four.
+
+# ## Caching
+
+# We can improve our formulation by caching the list of incoming and outgoing
+# nodes for each node `n`:
+
+out_nodes = Dict(n => Int[] for n in nodes)
+in_nodes = Dict(n => Int[] for n in nodes)
+for (i, j) in edges
+    push!(out_nodes[i], j)
+    push!(in_nodes[j], i)
+end
+
+# with the corresponding change to our model:
+
+model = Model()
+@variable(model, flows[e in edges] >= 0)
+@constraint(
+    model,
+    [n in nodes],
+    sum(flows[(i, n)] for i in in_nodes[n]) -
+    sum(flows[(n, j)] for j in out_nodes[n]) == demand(n)
+);
+
+# The benefit of this formulation is that we now loop over `out_nodes[n]`
+# rather than `edges` for each node `n`, and so the runtime is ``O(|edges|)``.
+
+# Let's build a new function to benchmark our formulation:
+
+function build_cached_model(nodes, edges, demand)
+    out_nodes = Dict(n => Int[] for n in nodes)
+    in_nodes = Dict(n => Int[] for n in nodes)
+    for (i, j) in edges
+        push!(out_nodes[i], j)
+        push!(in_nodes[j], i)
+    end
+    model = Model()
+    @variable(model, flows[e in edges] >= 0)
+    @constraint(
+        model,
+        [n in nodes],
+        sum(flows[(i, n)] for i in in_nodes[n]) -
+        sum(flows[(n, j)] for j in out_nodes[n]) == demand(n)
+    )
+    return model
+end
+
+nodes, edges, demand = build_random_graph(1_000, 2_000)
+@elapsed build_cached_model(nodes, edges, demand)
+
+# ## Analysis
+
+# Now we can analyse the difference in runtime of the two formulations:
+
+run_times_naive = Float64[]
+run_times_cached = Float64[]
+factors = 1:10
+for factor in factors
+    GC.gc()  #src
+    graph = build_random_graph(1_000 * factor, 5_000 * factor)
+    push!(run_times_naive, @elapsed build_naive_model(graph...))
+    push!(run_times_cached, @elapsed build_cached_model(graph...))
+end
+Plots.plot(; xlabel = "Factor", ylabel = "Runtime [s]")
+Plots.scatter!(factors, run_times_naive; label = "Actual")
+a, b = hcat(ones(10), factors .^ 2) \ run_times_naive
+Plots.plot!(factors, a .+ b * factors .^ 2; label = "Quadratic fit")
+Plots.scatter!(factors, run_times_cached; label = "Cached")
+a, b = hcat(ones(10), factors) \ run_times_cached
+Plots.plot!(factors, a .+ b * factors; label = "Linear fit")
+
+# Even though the cached model needs to build `in_nodes` and `out_nodes`, it is
+# asymptotically faster than the naïve model, scaling linearly with `factor`
+# rather than quadraticaly.
+
+# ## Lesson
+
+# If you write code with `sum-if` type conditions, for example,
+# `@constraint(model, [a in set], sum(x[b] for b in list if condition(a, b))`,
+# you can improve the performance by caching the elements for which `condition(a, b)`
+# is true.
+
+# Finally, you should understand that this behavior is not specific to JuMP, and
+# that it applies more generally to all computer programs you might write.
+# (Python programs that use Pyomo or gurobipy would similarly benefit from this
+# caching approach.)
+#
+# Understanding big-O notation and algorithmic complexity is a useful debugging
+# skill to have, regardless of the type of program that you are writing.

docs/styles/config/vocabularies/JuMP/accept.txt

@@ -42,6 +42,7 @@ preprint
 README
 recurse
 reimplemented
+runtime(?s)
 [Ss]tacktrace
 subexpression(?s)
 subgraph(?s)
@@ -136,6 +137,7 @@ Geomstats
 Geoopt
 GLPK
 (Gurobi|GUROBI)
+gurobipy
 GZip
 Hypatia
 (Ipopt|IPOPT)
@@ -157,6 +159,7 @@ Pavito
 Penbmi
 Penopt
 Plasmo
+Pyomo
 puzzlor
 RAPOSa
 Riemopt