Access one exact solution to linear matrix inequalities defined in an SDPA-Sparse-formatted file?

[PoeticExposition]

The function ClusteredLowRankSolver.sdpa_sparse_to_problem can be used to read an SDPA-sparse file and create a ClusteredLowRankSolver.Problem. In the SDPA format, the linear-matrix-inequality form is treated as the primal SDP, which means that in order to exactly solve the original problem stored in a .dat-s file, an exact dual solution to the corresponding ClusteredLowRankSolver.Problem is needed. However, it appears that ClusteredLowRankSolver.exact_solution can only return an exact primal optimal solution. As my primary goal is to obtain an exact solution to linear matrix inequalities (in which the complementary relaxation is, unfortunately, unlikely to be satisfied), I would like to know if ClusteredLowRankSolver.jl supports rounding the numerical dual solution as well?

[nanleij]

No, currently only the primal solution (from the equality form) can be rounded. Since ClusteredLowRankSolver supports free variables, it should be possible to create a constraints enforcing X = \sum_i y_i A_i - C with X PSD in the primal/equality formulation, but there currently is no function to produce the Problem corresponding to the LMI form of a sdpa-sparse file.

You can try to create the LMI form as a JuMP model, and then use the exact_solution function on the JuMP model (available since v2). I don't know how JuMP will handle that the conversion, but it might be that it will basically do what I wrote in the paragraph above, which would then give the possibility to round the solution you want. Maybe you need to add the matrices as PSD variables separately instead of just introducing the variables y_i and having a PSDcone constraint for the LMI's. That would give an SDP exactly in the form ClusteredLowRankSolver accepts.