Hello, I was wondering if there are algorithms for (linear) Semidefinite Programs (SDP) out there, that converge towards a maximally complementary solution, even if strict complementary does not hold. In particular: My SDP (for which I know stricly feasible starting points (primal and dual)) does not have any stricty complementary optimal solutions. Because of that I am interested in at least maximally complementary solutions. If yes, I would like to find a proof. Thanks for any help.
The standard approach is to embed the original SDP in a self dual formulation that has strictly feasible primal/dual solutions, solve the self dual formulation and then reach conclusions about the original problem from the solution of the self-dual problem.
See for example:
Unfortunately, I believe that this will only tell you when the problem has no strictly complementary primal-dual solutions and not necessarily give you a maximally complementary solution.