Superlinear and quadratic convergence of a stabilized sequential quadratic semidefinite programming method for nonlinear semidefinite programming problems

TL;DR

This paper introduces a stabilized sequential quadratic semidefinite programming method for nonlinear semidefinite programming, proving superlinear and quadratic local convergence under mild conditions, addressing a gap in existing methods.

Contribution

It develops a new stabilized SQSDP method with proven superlinear and quadratic convergence for NSDP problems, especially degenerate cases.

Findings

Proves superlinear convergence of the method.

Establishes quadratic convergence under mild assumptions.

Addresses a gap in stabilized SQSDP methods for NSDP.

Abstract

In this paper, we present a stabilized sequential quadratic semidefinite programming (SQSDP) method for nonlinear semidefinite programming (NSDP) problems and prove its local convergence. The stabilized SQSDP method is originally developed to solve degenerate NSDP problems and is based on the stabilized sequential programming (SQP) methods for nonlinear programming (NLP) problems. Although some SQP-type methods for NSDP problems have been proposed, most of them are SQSDP methods which are based on the SQP methods for NLP problems, and there are few researches regarding the stabilized SQSDP methods. In particular, there is room for the development of locally fast convergent stabilized SQSDP methods. We prove not only superlinear but also quadratic convergence of the proposed method under some mild assumptions, such as strict Robinson's constraint qualification and second-order sufficient condition.