Second-order sequential optimality conditions for nonlinear semidefinite optimization problems

TL;DR

This paper extends sequential optimality conditions to second-order for nonlinear semidefinite optimization, introducing AKKT2 and CAKKT2, which are necessary for local minima without constraint qualifications, and proposes an algorithm to find such points.

Contribution

It introduces the AKKT2 and CAKKT2 conditions for NSDP, providing necessary second-order optimality conditions without constraint qualifications and an algorithm to find points satisfying these conditions.

Findings

AKKT2 and CAKKT2 are necessary for local minima.

Under certain conditions, AKKT2 implies weak second-order necessary conditions.

A penalty-based algorithm generates sequences converging to points satisfying AKKT2 and CAKKT2.

Abstract

Sequential optimality conditions play an important role in constrained optimization since they provide necessary conditions without requiring constraint qualifications (CQs). This paper introduces a second-order extension of the Approximate Karush-Kuhn-Tucker (AKKT) conditions, referred to as AKKT2, for nonlinear semidefinite optimization problems (NSDP). In particular, we provide a formal definition of AKKT2, as well as its stronger variant, called Complementary AKKT2 (CAKKT2), and prove that these conditions are necessary for local minima without any assumption. Moreover, under Robinson's CQ and the weak constant rank property, we show that AKKT2 implies the so-called weak second-order necessary condition. Finally, we propose a penalty-based algorithm that generates sequences whose accumulation points satisfy the AKKT2 and the CAKKT2 conditions.