# American Institute of Mathematical Sciences

October  2011, 7(4): 977-989. doi: 10.3934/jimo.2011.7.977

## A smoothing homotopy method based on Robinson's normal equation for mixed complementarity problems

 1 School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, China 2 School of Mathematical Science, Dalian University of Technology, Dalian, Liaoning 116024, China

Received  January 2011 Revised  June 2011 Published  August 2011

In this paper, a probability-one homotopy method for solving mixed complementarity problems is proposed. The homotopy equation is constructed by using the Robinson's normal equation of mixed complementarity problem and a $C^2$-smooth approximation of projection function. Under the condition that the mixed complementarity problem has no solution at infinity, which is a weaker condition than several well-known ones, existence and convergence of a smooth homotopy path from almost any starting point in $\mathbb{R}^n$ are proven. The homotopy method is implemented in Matlab and numerical results on the MCPLIB test collection are given.
Citation: Zhengyong Zhou, Bo Yu. A smoothing homotopy method based on Robinson's normal equation for mixed complementarity problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 977-989. doi: 10.3934/jimo.2011.7.977
##### References:

show all references

##### References:
 [1] ShiChun Lv, Shou-Qiang Du. A new smoothing spectral conjugate gradient method for solving tensor complementarity problems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021150 [2] Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184 [3] Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196 [4] Kaili Zhang, Haibin Chen, Pengfei Zhao. A potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 429-443. doi: 10.3934/jimo.2018049 [5] Chunyang Zhang, Shugong Zhang, Qinghuai Liu. Homotopy method for a class of multiobjective optimization problems with equilibrium constraints. Journal of Industrial & Management Optimization, 2017, 13 (1) : 81-92. doi: 10.3934/jimo.2016005 [6] Xin Li, Feng Bao, Kyle Gallivan. A drift homotopy implicit particle filter method for nonlinear filtering problems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021097 [7] Li Chu, Bo Wang, Jie Zhang, Hong-Wei Zhang. Convergence analysis of a smoothing SAA method for a stochastic mathematical program with second-order cone complementarity constraints. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1863-1886. doi: 10.3934/jimo.2020050 [8] Xiao-Hong Liu, Wei-Zhe Gu. Smoothing Newton algorithm based on a regularized one-parametric class of smoothing functions for generalized complementarity problems over symmetric cones. Journal of Industrial & Management Optimization, 2010, 6 (2) : 363-380. doi: 10.3934/jimo.2010.6.363 [9] Jie-Wen He, Chi-Chon Lei, Chen-Yang Shi, Seak-Weng Vong. An inexact alternating direction method of multipliers for a kind of nonlinear complementarity problems. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 353-362. doi: 10.3934/naco.2020030 [10] Behrouz Kheirfam. A weighted-path-following method for symmetric cone linear complementarity problems. Numerical Algebra, Control & Optimization, 2014, 4 (2) : 141-150. doi: 10.3934/naco.2014.4.141 [11] Ming-Zheng Wang, M. Montaz Ali. Penalty-based SAA method of stochastic nonlinear complementarity problems. Journal of Industrial & Management Optimization, 2010, 6 (1) : 241-257. doi: 10.3934/jimo.2010.6.241 [12] Xiaofei Liu, Yong Wang. Weakening convergence conditions of a potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021080 [13] Wanbin Tong, Hongjin He, Chen Ling, Liqun Qi. A nonmonotone spectral projected gradient method for tensor eigenvalue complementarity problems. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 425-437. doi: 10.3934/naco.2020042 [14] Wei-Zhe Gu, Li-Yong Lu. The linear convergence of a derivative-free descent method for nonlinear complementarity problems. Journal of Industrial & Management Optimization, 2017, 13 (2) : 531-548. doi: 10.3934/jimo.2016030 [15] Hui-Qiang Ma, Nan-Jing Huang. Neural network smoothing approximation method for stochastic variational inequality problems. Journal of Industrial & Management Optimization, 2015, 11 (2) : 645-660. doi: 10.3934/jimo.2015.11.645 [16] Zheng-Hai Huang, Jie Sun. A smoothing Newton algorithm for mathematical programs with complementarity constraints. Journal of Industrial & Management Optimization, 2005, 1 (2) : 153-170. doi: 10.3934/jimo.2005.1.153 [17] Ya Li, ShouQiang Du, YuanYuan Chen. Modified spectral PRP conjugate gradient method for solving tensor eigenvalue complementarity problems. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020147 [18] Mingzheng Wang, M. Montaz Ali, Guihua Lin. Sample average approximation method for stochastic complementarity problems with applications to supply chain supernetworks. Journal of Industrial & Management Optimization, 2011, 7 (2) : 317-345. doi: 10.3934/jimo.2011.7.317 [19] Jie Zhang, Yue Wu, Liwei Zhang. A class of smoothing SAA methods for a stochastic linear complementarity problem. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 145-156. doi: 10.3934/naco.2012.2.145 [20] Yi Zhang, Liwei Zhang, Jia Wu. On the convergence properties of a smoothing approach for mathematical programs with symmetric cone complementarity constraints. Journal of Industrial & Management Optimization, 2018, 14 (3) : 981-1005. doi: 10.3934/jimo.2017086

2020 Impact Factor: 1.801