Interior point trajectories in semidefinite programming

Abstract

In this paper we study interior point trajectories in semidefinite programming (SDP) including the central path of an SDP. This work was inspired by the seminal work of Megiddo on linear programming trajectories [Progress in Math. Programming: Interior-Point Algorithms and Related Methods, N. Megiddo, ed., Springer-Verlag, Berlin, 1989, pp. 131-158]. Under an assumption of primal and dual strict feasibility, we show that the primal and dual central paths exist and converge to the analytic centers of the optimal faces of, respectively, the primal and the dual problems. We consider a class of trajectories that are similar to the central path but can be constructed to pass through any given interior feasible or infeasible point, and study their convergence. Finally, we study the derivatives of these trajectories and their convergence.