Local convergence of primal-dual interior point methods for nonlinear semidefinite optimization using the Monteiro-Tsuchiya family of search directions

TL;DR

This paper introduces a new primal-dual interior point method for nonlinear semidefinite optimization using Monteiro-Tsuchiya directions, proving local superlinear convergence under certain assumptions, expanding the scope beyond existing Monteiro-Zhang methods.

Contribution

It develops a primal-dual interior point method for NSDPs based on Monteiro-Tsuchiya directions, extending their application from linear to nonlinear problems and proving convergence properties.

Findings

Proposed a PDIPM with Monteiro-Tsuchiya directions for NSDPs.

Proved local superlinear convergence to KKT points.

Established convergence under general assumptions on scaling matrices.

Abstract

The recent advance of algorithms for nonlinear semi-definite optimization problems, called NSDPs, is remarkable. Yamashita et al. first proposed a primal-dual interior point method (PDIPM) for solving NSDPs using the family of Monteiro-Zhang (MZ) search directions. Since then, various kinds of PDIPMs have been proposed for NSDPs, but, as far as we know, all of them are based on the MZ family. In this paper, we present a PDIPM equipped with the family of Monteiro-Tsuchiya (MT) directions, which were originally devised for solving linear semi-definite optimization problems as were the MZ family. We further prove local superlinear convergence to a Karush-Kuhn-Tucker point of the NSDP in the presence of certain general assumptions on scaling matrices, which are used in producing the MT scaling directions.