A proximal point algorithm for the monotone second-order cone complementarity problem

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

This paper is devoted to the study of the proximal point algorithm for solving monotone second-order cone complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original problem. After given an appropriate criterion for approximate solutions of subproblems by adopting a merit function, the proximal point algorithm is verified to have global and superlinear convergence properties. For the purpose of solving the subproblems efficiently, we introduce a generalized Newton method and show that only one Newton step is eventually needed to obtain a desired approximate solution that approximately satisfies the appropriate criterion under mild conditions. Numerical comparisons are also made with the derivative-free descent method used by Pan and Chen (Optimization 59:1173-1197, 2010), which confirm the theoretical results and the effectiveness of the algorithm.

Original languageEnglish
Pages (from-to)1037-1063
Number of pages27
JournalComputational Optimization and Applications
Volume51
Issue number3
DOIs
Publication statusPublished - 2012 Apr 1

Fingerprint

Second-order Cone
Proximal Point Algorithm
Complementarity Problem
Cones
Monotone
Approximate Solution
Generalized Newton Method
Derivative-free Methods
Descent Method
Merit Function
Superlinear Convergence
Numerical Comparisons
Global Convergence
Convergence Properties
Regularization
Newton-Raphson method
Optimization
Derivatives

Keywords

  • Approximation criterion
  • Complementarity problem
  • Proximal point algorithm
  • Second-order cone

ASJC Scopus subject areas

  • Control and Optimization
  • Computational Mathematics
  • Applied Mathematics

Cite this

A proximal point algorithm for the monotone second-order cone complementarity problem. / Wu, Jia; Chen, Jein Shan.

In: Computational Optimization and Applications, Vol. 51, No. 3, 01.04.2012, p. 1037-1063.

Research output: Contribution to journalArticle

@article{c18a405e0e86407191d2b3b82cc102ce,
title = "A proximal point algorithm for the monotone second-order cone complementarity problem",
abstract = "This paper is devoted to the study of the proximal point algorithm for solving monotone second-order cone complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original problem. After given an appropriate criterion for approximate solutions of subproblems by adopting a merit function, the proximal point algorithm is verified to have global and superlinear convergence properties. For the purpose of solving the subproblems efficiently, we introduce a generalized Newton method and show that only one Newton step is eventually needed to obtain a desired approximate solution that approximately satisfies the appropriate criterion under mild conditions. Numerical comparisons are also made with the derivative-free descent method used by Pan and Chen (Optimization 59:1173-1197, 2010), which confirm the theoretical results and the effectiveness of the algorithm.",
keywords = "Approximation criterion, Complementarity problem, Proximal point algorithm, Second-order cone",
author = "Jia Wu and Chen, {Jein Shan}",
year = "2012",
month = "4",
day = "1",
doi = "10.1007/s10589-011-9399-x",
language = "English",
volume = "51",
pages = "1037--1063",
journal = "Computational Optimization and Applications",
issn = "0926-6003",
publisher = "Springer Netherlands",
number = "3",

}

TY - JOUR

T1 - A proximal point algorithm for the monotone second-order cone complementarity problem

AU - Wu, Jia

AU - Chen, Jein Shan

PY - 2012/4/1

Y1 - 2012/4/1

N2 - This paper is devoted to the study of the proximal point algorithm for solving monotone second-order cone complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original problem. After given an appropriate criterion for approximate solutions of subproblems by adopting a merit function, the proximal point algorithm is verified to have global and superlinear convergence properties. For the purpose of solving the subproblems efficiently, we introduce a generalized Newton method and show that only one Newton step is eventually needed to obtain a desired approximate solution that approximately satisfies the appropriate criterion under mild conditions. Numerical comparisons are also made with the derivative-free descent method used by Pan and Chen (Optimization 59:1173-1197, 2010), which confirm the theoretical results and the effectiveness of the algorithm.

AB - This paper is devoted to the study of the proximal point algorithm for solving monotone second-order cone complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original problem. After given an appropriate criterion for approximate solutions of subproblems by adopting a merit function, the proximal point algorithm is verified to have global and superlinear convergence properties. For the purpose of solving the subproblems efficiently, we introduce a generalized Newton method and show that only one Newton step is eventually needed to obtain a desired approximate solution that approximately satisfies the appropriate criterion under mild conditions. Numerical comparisons are also made with the derivative-free descent method used by Pan and Chen (Optimization 59:1173-1197, 2010), which confirm the theoretical results and the effectiveness of the algorithm.

KW - Approximation criterion

KW - Complementarity problem

KW - Proximal point algorithm

KW - Second-order cone

UR - http://www.scopus.com/inward/record.url?scp=84862017728&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84862017728&partnerID=8YFLogxK

U2 - 10.1007/s10589-011-9399-x

DO - 10.1007/s10589-011-9399-x

M3 - Article

AN - SCOPUS:84862017728

VL - 51

SP - 1037

EP - 1063

JO - Computational Optimization and Applications

JF - Computational Optimization and Applications

SN - 0926-6003

IS - 3

ER -