Complexity analysis of interior-point methods for second-order stationary points of nonlinear semidefinite optimization problems

TL;DR

This paper introduces a primal-dual interior-point method for nonlinear semidefinite optimization that guarantees convergence to second-order stationary points with a worst-case iteration complexity, advancing beyond first-order convergence.

Contribution

The paper presents the first primal-dual interior-point method for NSDPs that converges to SOSPs with proven iteration complexity, incorporating directions of negative curvature.

Findings

Method converges to SOSPs with worst-case iteration complexity.

Using negative curvature directions improves convergence behavior.

Numerical experiments demonstrate practical benefits of the proposed approach.

Abstract

We propose a primal-dual interior-point method (IPM) with convergence to second-order stationary points (SOSPs) of nonlinear semidefinite optimization problems, abbreviated as NSDPs. As far as we know, the current algorithms for NSDPs only ensure convergence to first-order stationary points such as Karush-Kuhn-Tucker points, but without a worst-case iteration complexity. The proposed method generates a sequence approximating SOSPs while minimizing a primal-dual merit function for NSDPs by using scaled gradient directions and directions of negative curvature. Under some assumptions, the generated sequence accumulates at an SOSP with a worst-case iteration complexity. This result is also obtained for a primal IPM with a slight modification. Finally, our numerical experiments show the benefits of using directions of negative curvature in the proposed method.