Primal-Dual Path-Following Algorithms for Semidefinite Programming

This paper deals with a class of primal-dual interior-point algorithms for semideenite programming (SDP) which was recently introduced by Kojima, Shindoh and Hara 11]. These authors proposed a family of primal-dual search directions that generalizes the one used in algorithms for linear programming based on the scaling matrix X 1=2 S ?1=2. They study three primal-dual algorithms based on this family of search directions: a short-step path-following method, a feasible potential-reduction method and an infeasible potential-reduction method. However, they were not able to provide an algorithm which generalizes the long-step path-following algorithm introduced by Kojima, Mizuno and Yoshise 10]. In this paper, we characterize two search directions within their family as being (unique) solutions of systems of linear equations in symmetric variables. Based on this characterization, we present: 1) a simpliied polynomial convergence proof for one of their short-step path-following algorithm and, 2) for the rst time, a polynomially convergent long-step path-following algorithm for SDP which requires an extra p n factor in its iteration-complexity order as compared to its linear programming counterpart, where n is the number of rows (or columns) of the matrices involved.