Support for skew-symmetric matrix variables

[frederikgeth]

I contribute to PowerModelsDistribution. Ideally, we would like to use JuMP's support for matrix variables with symmetry more broadly.

As we start from a complex-value problem with Hermitian matrix variables, we would love to see support for both symmetric (real part) and skew-symmetric (imaginary part) matrix variables. We have a work-around, but the printing of the variables looks different from other variables.

We can do something like the following to make sets of full and symmetric matrix variables:

Dict(i => JuMP.@variable(pm.model,
            [c in 1:N, d in 1:N], base_name="P_$(i)",
        ) for i in ids  )
Dict(i => JuMP.@variable(pm.model,
            [c in 1:N, d in 1:N], base_name="Wr_$(i)", Symmetric
        ) for i in ids  )

but the following is not supported:

Dict(i => JuMP.@variable(pm.model,
            [c in 1:N, d in 1:N], base_name="Wi_$(i)", SkewSymmetric
        ) for i in ids  )

Ideally, this would have (support for) zeros on the diagonal as well.

After some preliminary digging, it seems Julia does not have skew symmetric matrix variable primitives (https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/#Special-matrices-1, somewhat related discussion in JuliaLang/LinearAlgebra.jl#417)

[blegat]

The proper way would be to define a SkewMatrixSpace similar to
https://github.com/JuliaOpt/JuMP.jl/blob/master/src/sd.jl#L1
and a SkewMatrixShape similar to
https://github.com/JuliaOpt/JuMP.jl/blob/master/src/sd.jl#L76
and a build_variable similar to
https://github.com/JuliaOpt/JuMP.jl/blob/master/src/sd.jl#L163
So that the user can do

@variable(model, x[1:3, 1:3] in SkewMatrixSpace()) 

[frederikgeth]

Thanks, @blegat, those pointers helped a lot. I found a way to make it work in PR #2147.