Add HermitianPositiveSemidefiniteConeTriangle with bridges

[blegat]

Refactored from https://github.com/jump-dev/ComplexOptInterface.jl

[odow]

I didn't dig into the reformulation. But I don't think we want to rush this into MOI v1.7. It's quite complicated.

[blegat]

Ok, I'll merge #1959 then

[chriscoey]

This PR should allow me to add direct support for this MOI Hermitian PSD cone to Hypatia's MOI interface

[odow]

Actually, just hold off. I'm tidying the docstrings:

https://jump.dev/MathOptInterface.jl/previews/PR1962/submodules/Bridges/list_of_bridges/#MathOptInterface.Bridges.Variable.HermitianToSymmetricPSDBridge

[odow]

@blegat take another look. I fleshed out the docstring with an example of the reformulation. I think it makes things a bit clearer.

I also removed your note on duality. It was a bit confusing and out of place. Once you see the example of the reformulation it becomes obvious why you can just sum up the elements. The equality constraints are just enforcing parts of the matrix, not anything structural.

[blegat]

Once you see the example of the reformulation it becomes obvious why you can just sum up the elements. The equality constraints are just enforcing parts of the matrix, not anything structural.

Why is that obvious that you don't have to include the dual of ceq ? The way I convince myself is as follows:
For the entry x_{11}, we have P_{11} - P_{'44} as constraints so if \lambda is the dual of this constraint the dual at 11 gets \lambda and at 44, it gets -\lambda. But the dual obtained by multiplying by A* so it's the sum of the dual at 11 + the dual at 44 so you get \lambda - \lambda = 0.
Is there an easier way to get this ?

[odow]

I didn't try to prove haha. It just seemed right. We don't actually implement the reformulated PSD matrix, so in some sense, the equality constraints are a lower-level implementation detail. What we actually ask for is the duals on that PSD matrix.

[odow]

@blegat, some solvers are failing this test: https://github.com/jump-dev/SolverTests/actions/runs/2857418874

Is it possible for the dual to be non-zero?