Superlinear Convergence of a Symmetric Primal-Dual Path Following Algorithm for Semidefinite Programming

This paper establishes the superlinear convergence of a symmetric primal dual path following algorithm for semide nite programming under the assumptions that the semide nite pro gram has a strictly complementary primal dual optimal solution and that the size of the central path neighborhood tends to zero The interior point algorithm considered here closely resembles the Mizuno Todd Ye predictor correctormethod for linear programmingwhere it is known to be quadrat ically convergent It is shown that when the iterates are well centered the duality gap is reduced superlinearly after each predictor step Indeed if each predictor step is succeeded by r consecutive corrector steps then the predictor reduces the duality gap superlinearlywith order r The proof relies on a careful analysis of the central path for semide nite programming It is shown that under the strict complementarity assumption the primal dual central path converges to the analytic center of the primal dual optimal solution set and the distance from any point on the central path to this analytic center is bounded by the duality gap