Analysis of the primal-dual central path for nonlinear semidefinite optimization without the nondegeneracy condition

TL;DR

This paper investigates the properties of the central path in nonlinear semidefinite optimization without relying on the nondegeneracy condition, establishing existence, uniqueness, and convergence results under weaker assumptions.

Contribution

It extends the understanding of the primal-dual central path by removing the nondegeneracy condition, proving existence, uniqueness, and convergence properties under milder assumptions.

Findings

The central path exists uniquely without the nondegeneracy condition.

The dual component converges to the analytic center of the Lagrange multiplier set.

A region is identified where Newton's equations are uniquely solvable.

Abstract

We study properties of the central path underlying a nonlinear semidefinite optimization problem, called NSDP for short. The latest radical work on this topic was contributed by Yamashita and Yabe (2012): they proved that the Jacobian of a certain equation-system derived from the Karush-Kuhn-Tucker (KKT) conditions of the NSDP is nonsingular at a KKT point under the second-order sufficient condition (SOSC), the strict complementarity condition (SC), and the nondegeneracy condition (NC). This yields uniqueness and existence of the central path through the implicit function theorem. In this paper, we consider the following three assumptions on a KKT point: the strong SOSC, the SC, and the Mangasarian-Fromovitz constraint qualification. Under the absence of the NC, the Lagrange multiplier set is not necessarily a singleton and the nonsingularity of the above-mentioned Jacobian is no longer valid. Nonetheless, we establish that the central path exists uniquely, and moreover prove that the dual component of the path converges to the so-called analytic center of the Lagrange multiplier set. As another notable result, we clarify a region around the central path where Newton's equations relevant to primal-dual interior point methods are uniquely solvable.