A revised sequential quadratic semidefinite programming method for nonlinear semidefinite optimization

TL;DR

This paper introduces a revised sequential quadratic semidefinite programming method that guarantees convergence to complementarity-AKKT points in nonlinear semidefinite optimization, enhancing previous algorithms with practical convergence properties.

Contribution

The paper proposes a modified SQSDP algorithm that ensures global convergence to CAKKT points, extending the original method to more practical optimality conditions.

Findings

Proved global convergence to CAKKT points.

Maintained desirable properties of the original method.

Presented preliminary numerical results.

Abstract

In 2020, Yamakawa and Okuno proposed a stabilized sequential quadratic semidefinite programming (SQSDP) method for solving, in particular, degenerate nonlinear semidefinite optimization problems. The algorithm is shown to converge globally without a constraint qualification, and it has some nice properties, including the feasible subproblems, and their possible inexact computations. In particular, the convergence was established for approximate-Karush-Kuhn-Tucker (AKKT) and trace-AKKT conditions, which are two sequential optimality conditions for the nonlinear conic contexts. However, recently, complementarity-AKKT (CAKKT) conditions were also consider, as an alternative to the previous mentioned ones, that is more practical. Since few methods are shown to converge to CAKKT points, at least in conic optimization, and to complete the study associated to the SQSDP, here we propose a revised version of the method, maintaining the good properties. We modify the previous algorithm, prove the global convergence in the sense of CAKKT, and show some preliminary numerical experiments.