Optimality Conditions for Nonlinear Semidefinite Programming via Squared Slack Variables

TL;DR

This paper develops second-order optimality conditions for nonlinear semidefinite programming by reformulating the problem with squared slack variables, providing a new perspective on optimality and regularity conditions.

Contribution

It introduces a reformulation of NSDP using squared slack variables and establishes equivalent second-order optimality conditions with potential computational benefits.

Findings

Derived second-order optimality conditions for NSDP

Established correspondence between KKT points and regularity conditions

Analyzed computational prospects of the squared slack variables approach

Abstract

In this work, we derive second-order optimality conditions for nonlinear semidefinite programming (NSDP) problems, by reformulating it as an ordinary nonlinear programming problem using squared slack variables. We first consider the correspondence between Karush-Kuhn-Tucker points and regularity conditions for the general NSDP and its reformulation via slack variables. Then, we obtain a pair of "no-gap" second-order optimality conditions that are essentially equivalent to the ones already considered in the literature. We conclude with the analysis of some computational prospects of the squared slack variables approach for NSDP.