support complex PSD cone

[araujoms]

This adds the Hermitian bridge, which seems to do its job, but then the code dies when it reaches optimize! with the error "MathOptInterface.VectorAffineFunction{ComplexF64}-in-SeDuMi.ScaledPSDCone constraint is not supported by the model."

I suppose I need to edit OptimizerCache to tell MOI that this is actually supported, but I don't know how.

Closes #23

[araujoms]

Sorry about the delay. Now the basic solving is working. Two problems remain: I can't figure out how to get extraction of dual variables to work, and the MOI tests are failing. The test failures seem to be due to MOI assuming that the raw solution returned by the solver is always real. This is not the case with SeDuMi, so if I understood it correctly the fix should be done there.

[odow]

I'm not in a position to competently review this because I don't have MATLAB to test.

[araujoms]

I've made some progress in returning the dual variables; at least now the algorithm is in place, the remaining issue is calling it correctly. As you can see in lines 253-254, it works correctly if I manually switch between asking for the dual variables for the real and complex cones, but not both at the same time.

Also, I'm confused about the purpose of the function MOI.Bridges.inverse_map_function. Shouldn't it just be the same as MOI.Bridges.adjoint_map_function?

[araujoms]

@blegat can you help me here?

[blegat]

Thanks for the reminder, I'll take a look

[blegat]

Maybe it's easier to instead add a bridge from MOI.VectorAffineFunction{T}-in-MOI.HermitianPositiveSemidefiniteConeTriangle to MOI.VectorAffineFunction{Complex{T}}-in-MOI.HermitianPositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{T}

[blegat]

@araujoms Can you try removing ScaledHermitianPSDConeBridge and replacing it with jump-dev/MathOptInterface.jl#2724 ?

[araujoms]

I tried, got a bunch of errors. Why, though? Is this to get the MOI tests to pass? Solving was working correctly before, the part that was missing was recovering dual variables.

What I did is removing everything related to ScaledHermitianPSDConeBridge, and replacing the function that returns the list of nonstandard bridges with

function MOI.get(::Optimizer, ::MOI.Bridges.ListOfNonstandardBridges)
    return Type[
        ScaledPSDConeBridge{Float64},
        MOI.Bridges.Constraint.HermitianToComplexSymmetricBridge{Float64},
    ]
end

[blegat]

Why, though?

I prefer keeping simple bridges that do just one transformation rather than complicated bridges that mixes several ones. Especially when you need to figure out what's the adjoint of the inverse of the adjoint of the combination of several transformations. Here one of the two transformations I'm mentioning is the bridge from the MOI set to the SeDuMi set which is solver-specific and the other transformation is the one of jump-dev/MathOptInterface.jl#2724.
That PR is still a work in progress. Just pushed some fixes, only the adjoint and inverse adjoint are missing now

[blegat]

I have implemented the adjoint, can you try now ?

[araujoms]

Sure. Same error as before:

ERROR: MethodError: no method matching promote_operation(::typeof(*), ::Type{Float64}, ::Type{Float64}, ::Type{MathOptInterface.ScalarAffineFunction{ComplexF64}})
The function `promote_operation` exists, but no method is defined for this combination of argument types.

I don't understand how is your new bridge supposed to work, though. ScaledHermitianPSDConeBridge mapped HermitianPositiveSemidefiniteConeTriangle to ScaledPSDCone, but your bridge is mapping it to PositiveSemidefiniteConeTriangle.

[blegat]

Yes but then ScaledPSDConeBridge can map VAF{Complex{Float64}}-in-PositiveSemidefiniteConeTriangle to VAF{Complex{Float64}}-in-ScaledPSDCone.

[blegat]

Ok I fixed that error on the MOI PR. I suspect we may still need to fix ScaledPSDConeBridge so that it works with a function of complex coefficients

[araujoms]

Now it doesn't produce any errors, but it doesn't use SeDuMi's complex PSD cone, but the real one:

 * Supported objective: MOI.ScalarAffineFunction{Float64}
 * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Zeros
 * Unsupported constraint: MOI.VectorOfVariables-in-MOI.HermitianPositiveSemidefiniteConeTriangle
 |  bridged by:
 |   MOIB.Constraint.HermitianToSymmetricPSDBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.VectorOfVariables}
 |  may introduce:
 |   * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeTriangle
 |   |  bridged by:
 |   |   SeDuMi.ScaledPSDConeBridge{Float64, MOI.VectorAffineFunction{Float64}}
 |   |  may introduce:
 |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-SeDuMi.ScaledPSDCone

In order for ScaledPSDConeBridge to accept complex numbers, perhaps we need to change something here:

struct ScaledPSDConeBridge{T,G} <: MOI.Bridges.Constraint.SetMapBridge{
    T,
    ScaledPSDCone,
    MOI.PositiveSemidefiniteConeTriangle,
    MOI.VectorAffineFunction{T},
    G,
}
    constraint::MOI.ConstraintIndex{MOI.VectorAffineFunction{T},ScaledPSDCone}
end

Is T somehow restricted to be real by MOI?

[blegat]

You may need to add ScaledPSDConeBridge{Complex{Float64}} here

[araujoms]

No luck, still get the same error as before.

[blegat]

Can you push your work in progress that gets this error as a commit to this branch ?

[araujoms]

sure, done

[odow]

I think the problem is that it expects optimizer.sol.x to be real, but it is in general complex with SeDuMi.

MOI doesn't care what the internal type is. You just need to make sure that the return types from MOI.get are what it expects.

[odow]

p.s., I've asked MATLAB for a license, because it's virtually impossible for me to review this PR without CI.

[araujoms]

That was it! MOI tests pass now. I only had to do the analogous change to querying the slack variables.

[blegat]

Awesome, the PR is still blocked by

• Fix SetMapBridge with complex type MathOptInterface.jl#2733
• A new release of MOI

[araujoms]

Just tested with MOI 1.40, everything works fine.

[odow]

Please remove these lines and use MOI.VectorAffineFunction{Float64} and MOI.VectorAffineFunction{ComplexF64} everywhere.

[araujoms]

I doesn't even compile if I do that. In any case this is Benoît's code, not mine.

[blegat]

I added a comment above explaining why they are needed

[blegat]

Tests are passing

shell> git log HEAD^..HEAD
commit 637db910cc5a28f8d8fed0655cc184c86210ee2d (HEAD -> complex, araujoms/complex)
Author: Benoît Legat <[email protected]>
Date:   Tue May 6 08:47:27 2025 +0200

    Fix format

julia> versioninfo()
Julia Version 1.10.2
Commit bd47eca2c8a (2024-03-01 10:14 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 36 × Intel(R) Xeon(R) W-2295 CPU @ 3.00GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, cascadelake)
Threads: 1 default, 0 interactive, 1 GC (on 36 virtual cores)
Environment:
  JULIA_EDITOR = code
  JULIA_NUM_THREADS = 

julia> include("test/runtests.jl");
Precompiling SeDuMi
  1 dependency successfully precompiled in 2 seconds. 39 already precompiled.
Test Summary:              | Pass  Total  Time
Linear Programming example |    7      7  2.1s
Test Summary:                    | Pass  Total  Time
Semidefinite Programming example |   10     10  0.6s
>> SeDuMi 1.3.7 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 1, order n = 3, dim = 9, blocks = 2
nnz(A) = 1 + 0, nnz(ADA) = 1, nnz(L) = 1
 it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
  0 :            8.33E+00 0.000
  1 :   5.15E-01 1.60E+00 0.000 0.1923 0.9000 0.9000   1.46  1  1  7.6E-01
  2 :   9.95E-01 4.27E-02 0.000 0.0267 0.9900 0.9900   0.95  1  1  1.9E-02
  3 :   1.00E+00 4.38E-07 0.000 0.0000 1.0000 1.0000   1.00  1  1  1.6E-07
  4 :   1.00E+00 4.54E-14 0.000 0.0000 1.0000 1.0000   1.00  1  1  1.6E-14

iter seconds digits       c*x               b*y
  4      0.1  14.1  1.0000000000e+00  1.0000000000e+00
|Ax-b| =   5.4e-15, [Ay-c]_+ =   0.0E+00, |x|=  1.0e+00, |y|=  1.0e+00

Detailed timing (sec)
   Pre          IPM          Post
3.634E-02    5.408E-02    2.404E-03    
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.
Test Summary: | Pass  Total  Time
test_complex  |    4      4  3.5s
Test Summary:        | Pass  Total  Time
test_complex_bridges |    4      4  3.5s
Test Summary: | Pass  Total  Time
test_options  |    1      1  0.0s
>> Warning: Rank deficient, rank = 1, tol =  4.615110e-14. 
> In sedumi (line 272)
 
>> Warning: Rank deficient, rank = 1, tol =  4.615110e-14. 
> In sedumi (line 272)
 
Test Summary: | Pass  Total     Time
test_runtests | 4070   4070  5m05.0s
Test Summary:    | Pass  Total  Time
test_solver_name |    1      1  0.0s

[blegat]

@araujoms Thanks a lot !!

[araujoms]

Thank you! That was quite an odyssey.