Asymptotic behavior of the central path for a special class of degenerate SDP problems

This paper studies the asymptotic behavior of the central path (X(ν), S(ν), y(ν)) as ν ↓ 0 for a class of degenerate semidefinite programming (SDP) problems, namely those that do not have strictly complementary primal-dual optimal solutions and whose “degenerate diagonal blocks” XT (ν) and ST (ν) of the central path are assumed to satisfy max{‖XT (ν)‖, ‖ST (ν)‖} = O(√ν). We establish the convergence of the central path towards a primal-dual optimal solution, which is characterized as being the unique optimal solution of a certain log-barrier problem. A characterization of the class of SDP problems which satisfy our assumptions are also provided. It is shown that the re-parametrization t > 0 → (X(t4), S(t4), y(t4)) of the central path is analytic at t = 0. The limiting behavior of the derivative of the central path is also investigated and it is shown that the order of convergence of the central path towards its limit point is O( √ ν). Finally, we apply our results to the convex quadratically constrained convex programming (CQCCP) problem and characterize the class of CQCCP problems which can be formulated as SDPs satisfying the assumptions of this paper. In particular, we show that CQCCP problems with either a strictly convex objective function or at least one strictly convex constraint function lie in this class.