Sensitivity part II

[odow]

Closes #2244

This PR is a rewrite of the sensitivity summary to avoid a lot of redundant computations and allocations.

I've tried to comment my understanding of the code, and compared against the solutions reported by Gurobi and Excel.

Points of contention are:

• What is the sensitivity of a RHS of an Interval constraint? I have left unimplemented.
• What is the objective sensitivity of a nonbasic variable with fixed bounds? Gurobi seems to report +/-0.0, where as Excel reports +/-Inf. The current behavior is +/-Inf.

I'm not sure I fully understand all of the edge cases, so it will be interesting to see what the coverage is.

cc @geoffleyland: Want to stress-test this?

[codecov]

Codecov Report

Merging #2251 (5c47a44) into master (508d3a5) will increase coverage by 0.24%.
The diff coverage is 99.31%.

@@            Coverage Diff             @@
##           master    #2251      +/-   ##
==========================================
+ Coverage   91.61%   91.86%   +0.24%     
==========================================
  Files          42       43       +1     
  Lines        4451     4596     +145     
==========================================
+ Hits         4078     4222     +144     
- Misses        373      374       +1     

Impacted Files	Coverage Δ
src/JuMP.jl	74.59% <ø> (ø)
src/lp_sensitivity2.jl	99.30% <99.30%> (ø)
src/lp_sensitivity.jl	93.56% <100.00%> (+0.07%)	⬆️

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[joaquimg]

matrix building is really similar to what is done in:
https://github.com/JuliaOpt/MatrixOptInterface.jl

Matrix building (from MOI) will also be used in the differentiation GSOC.
A more general version of that is replicated in solvers like SCS and ProxSDP.

I think the matrix builder should live in MOIU. Starting with LP, for this use, then do QP (for GSOC) and Conic (also for GSOC, but potentially used in solvers).

This could simplify some first version solver wrappers.
And could be used by students writing their first solvers.

[odow]

👍 to adding matrix-type support to MOI as a submodule.

[joaquimg]

I have seen that Xpress allows you to query rhs and obj sensitivities.
Also, Gurobi and Xpress allow you to pass a vector and they will multiply it by the inverse basis matrix.

[odow]

Yeah some solvers do allow you to query the sensitivity. But I'm not sure how that plays with bridges, etc.

[joaquimg]

Indeed, that is a bit of a pain

[odow]

Anyone interested in reviewing this? If not, I'll merge in a few days. It's a significant improvement over what we have currently, which is unusable for anything but toy models.

[odow]

Bump. Anyone still want to review this/objections to merging?