Preprocessing and Reduction for Degenerate Semidefinite Programs

4 This paper presents a backward stable preprocessing technique for (nearly) ill-posed semidef5 inite programming, SDP, problems, i.e., programs for which Slater’s constraint qualification, 6 existence of strictly feasible points, (nearly) fails. 7 Current popular algorithms for semidefinite programming rely on primal-dual interior-point, 8 p-d i-p methods. These algorithms require Slater’s constraint qualification for both the primal 9 and dual problems. This assumption guarantees the existence of Lagrange multipliers, well10 posedness of the problem, and stability of algorithms. However, there are many instances of 11 SDPs where Slater’s constraint qualification fails or nearly fails. Our backward stable prepro12 cessing technique is based on finding a rank-revealing rotation of the problem followed by facial 13 reduction. This results in a smaller, well-posed, nearby problem that can be solved by standard 14 SDP solvers. 15