### 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 language | English |
---|---|

Pages (from-to) | 1037-1063 |

Number of pages | 27 |

Journal | Computational Optimization and Applications |

Volume | 51 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2012 Apr 1 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*Computational Optimization and Applications*, vol. 51, no. 3, pp. 1037-1063. https://doi.org/10.1007/s10589-011-9399-x

}

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 -