More optimal 0D positivity constraint?

[ianan]

Current python code is based on idl map approach where it finds (in default mode considering the positivity constrain) the regularization parameter λ from a range of values of λ, the one that satisfies

[1]. ||Kξ_λ -g||² - ρ||δg||² = min

But if ξ_λ < 0 repeats the above condition, increasing ρ, again looking for the λ that satisfies it, doing this iteratively until ξ_λ > 0 or max_iter is reached. This is the 'faster' approach used in the maps as only have to compute one ξ_λ and check its positivity once found the λ that satisfies [1], since [1] computable directly from GSVD output (i.e. don't need to compute ξ_λ to get [1].).

Would it be more optimal (avoiding while loop and ρ tweaking) to instead compute ξ_λ for a larger sample of λ values and then for the subset that satisfy ξ_λ > 0 only then do the [1]. condition. Likely would find higher minimum, which is equivalent to increasingly ρ in current approach. This newer version should find the best solution but might be slower to compute (i.e. bad for maps) since having to do lots of ξ_λ and then [1].

Shouldn't be as 'slow; as original approach in idl_org pos where [1]. and ξ_λ where both calculate for a large sample of λ and then λ satisfying the joint criteria was found.