Interior Point Trajectories in Semidefinite Programming

Don Goldfarb and Katya Scheinberg

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 by Megiddo on linear programming trajectories \cite{M}. 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 point and study their convergence. Finally, we study the first order derivatives of these trajectories and their convergence. We also consider higher order derivatives associated with these trajectories.

Technical Report, Department of IE/OR, Columbia University, New York, March, 1996. Revised November, 1996.