A stabilized sequential quadratic semidefinite programming method for degenerate nonlinear semidefinite programs

TL;DR

This paper introduces a new stabilized sequential quadratic semidefinite programming method for degenerate nonlinear semidefinite programs, allowing inexact solutions and removing the need for certain constraint qualifications, with proven global convergence.

Contribution

The paper presents a novel SQSDP method that solves QSDP subproblems inexactly and guarantees global convergence without standard constraint qualifications.

Findings

Method converges globally to stationary or AKKT points.

Allows inexact solutions of QSDP subproblems.

Numerical experiments demonstrate efficiency.

Abstract

In this paper, we propose a new sequential quadratic semidefinite programming (SQSDP) method for solving degenerate nonlinear semidefinite programs (NSDPs), in which we produce iteration points by solving a sequence of stabilized quadratic semidefinite programming (QSDP) subproblems, which we derive from the minimax problem associated with the NSDP. Unlike the existing SQSDP methods, the proposed one allows us to solve those QSDP subproblems inexactly, and each QSDP is feasible. One more remarkable point of the proposed method is that constraint qualifications (CQs) or boundedness of Lagrange multiplier sequences are not required in the global convergence analysis. Specifically, without assuming such conditions, we prove the global convergence to a point satisfying any of the following: the stationary conditions for the feasibility problem, the approximate-Karush-Kuhn-Tucker (AKKT) conditions, and the trace-AKKT conditions. Finally, we conduct some numerical experiments to examine the efficiency of the proposed method.