add relative entropy cone

[chriscoey]

This 1+2n-dimensional cone is useful for entropy optimization and has applications in e.g. signomial positivity certificates (see here and here). I've implemented a bridge to n exponential cone (3-dim) constraints and 1 LP constraint.

Hypatia supports this cone due to what is claimed to be a self-concordant barrier on p5 of "Primal-Dual Interior-Point Methods for Domain-Driven Formulations" by Karimi & Tuncel, 2019 (they didn't provide a SC proof there but I'd be interested to see one).

[chriscoey]

Let me guess... I need to redo the bridge as a set map bridge?

[blegat]

Let me guess... I need to redo the bridge as a set map bridge?

No, you create extra variables and more than one constraint so it's not a set map bridge.

[chriscoey]

Understood, thanks!

[chriscoey]

I'll attempt the starting values here next

[codecov-io]

Codecov Report

Merging #993 into master will decrease coverage by 0.01%.
The diff coverage is 93.06%.

@@            Coverage Diff             @@
##           master     #993      +/-   ##
==========================================
- Coverage   95.21%   95.19%   -0.02%     
==========================================
  Files          98       99       +1     
  Lines       11048    11149     +101     
==========================================
+ Hits        10519    10613      +94     
- Misses        529      536       +7

Impacted Files	Coverage Δ
src/Bridges/Constraint/norm_to_lp.jl	91.48% <ø> (ø)	⬆️
src/Utilities/model.jl	94.87% <ø> (ø)	⬆️
src/FileFormats/MOF/MOF.jl	100% <ø> (ø)	⬆️
src/Test/UnitTests/basic_constraint_tests.jl	98.21% <ø> (ø)	⬆️
src/FileFormats/MOF/write.jl	65.59% <0%> (-0.72%)	⬇️
src/Bridges/Constraint/Constraint.jl	96.55% <100%> (+0.12%)	⬆️
src/FileFormats/MOF/read.jl	93.16% <100%> (+0.05%)	⬆️
src/sets.jl	96.59% <100%> (+0.03%)	⬆️
src/Test/contconic.jl	99.24% <100%> (+0.01%)	⬆️
src/Bridges/Constraint/relentr_to_exp.jl	91.3% <91.3%> (ø)
... and 1 more

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[chriscoey]

OK so I've done my best to implement starting values. AFAICT it requires the Lambert W function. This is likely to be needed in similar contexts for future cones to be added to MOI that use Bridges with exponential cone extended formulations. The Lambert W function comes from the registered/up-to-date https://github.com/jlapeyre/LambertW.jl package, which is a small dependency I've added.

[chriscoey]

@blegat this is ready for review

[chriscoey]

Note btw that relative entropy is the perspective of negative log, so this cone can also be used for modeling a sum of logs, and hence is useful for a natural reformulation of geometric programming problems.

[odow]

So LambertW.jl is only one file, and under MIT, so potentially we could just include that file in MOI.Utilities? It hasn't been updated in > 1 year, and its Project.toml file is out-of-date.

[chriscoey]

@blegat's solution is better than mine so I changed that. thanks! no more lambertW