Preprocessing and Regularization for Degenerate Semidefinite Programs

This paper presents a backward stable preprocessing technique for (nearly) ill-posed semidef5 inite programming, SDP, problems, i.e., programs for which the Slater 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 the Slater constraint qualification for both the 9 primal and dual problems. This assumption guarantees the existence of Lagrange multipliers, 10 well-posedness of the problem, and stability of algorithms. However, there are many instances 11 of SDPs where the Slater constraint qualification fails or nearly fails. Our backward stable 12 preprocessing technique is based on applying the Borwein-Wolkowicz facial reduction process 13 to find a finite number, k, of rank-revealing orthogonal rotations of the problem. After an 14 appropriate truncation, this results in a smaller, well-posed, nearby problem that satisfies the 15 Robinson constraint qualification, and one that can be solved by standard SDP solvers. The 16 case k = 1 is of particular interest and is characterized by strict complementarity of an auxiliary 17 problem. 18