A simple proof of second-order sufficient optimality conditions in nonlinear semidefinite optimization

TL;DR

This paper provides an elementary proof of second-order sufficient optimality conditions in nonlinear semidefinite optimization, avoiding complex second-order tangent theory by explicitly computing the second subderivative.

Contribution

It introduces a straightforward proof method based on explicit computation of the second subderivative, simplifying existing theoretical approaches.

Findings

Simplifies the proof of second-order optimality conditions

Uses explicit computation of second subderivative

Recovers known curvature terms in the literature

Abstract

In this note, we present an elementary proof for a well-known second-order sufficient optimality condition in nonlinear semidefinite optimization which does not rely on the enhanced theory of second-order tangents. Our approach builds on an explicit elementary computation of the so-called second subderivative of the indicator function associated with the semidefinite cone which recovers the best curvature term known in the literature.