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

Takayuki Okuno

TL;DR

This paper presents a PDIPM equipped with the family of Monteiro–Tsuchiya (MT) directions, which were originally devised for solving linear semidefinite optimization problems as were the MZ family and proves local superlinear convergence to a Karush–Kuhn–Tucker point of the NSDP.

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.

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

TL;DR

This paper presents a PDIPM equipped with the family of Monteiro–Tsuchiya (MT) directions, which were originally devised for solving linear semidefinite optimization problems as were the MZ family and proves local superlinear convergence to a Karush–Kuhn–Tucker point of the NSDP.

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.

Paper Structure

This paper contains 22 sections, 11 theorems, 136 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The KKT and BKKT optimality conditions
  4. The MZ family of search directions for the NSDP
  5. The MT family of search directions for the NSDP
  6. Motivation of using the MT family
  7. Primal-dual interior point method based on the MT family of search directions
  8. Description of the proposed algorithm
  9. Globalization of Algorithm \ref{['alg4']}
  10. Some remarks on solving Newton equations
  11. Local convergence analysis of Algorithm \ref{['alg4']}
  12. Main results
  13. Preliminary results for Theorem \ref{['thm:main']}
  14. Uniqueness and error bounds of the solutions of the linear equations \ref{['eq:pre3']} and \ref{['eq:corr4']}
  15. Effectiveness of tangential and centering steps
  16. ...and 7 more sections

Key Result

proposition 1

Let $\{\tilde{\mu}_{\ell}\}\subseteq \mathbb{R}_{++}$ and $\{r_{\ell}\}\subseteq \mathbb{R}_{++}$ be a sequence such that $\lim_{\ell\to\infty}\tilde{\mu}_{\ell}=0$ and $r_{\ell}=o(\tilde{\mu}_{\ell})$. Moreover, let $\{\tilde{w}^{\ell}\}\subseteq \mathcal{W}_{++}$ be a sequence satisfying $\tilde{w

Theorems & Definitions (23)

  • definition 1
  • definition 2
  • remark 1
  • definition 3
  • definition 4
  • remark 2
  • proposition 1
  • proposition 2
  • theorem 1
  • proposition 3
  • ...and 13 more