American Institute of Mathematical Sciences

Advanced
April  2010, 6(2): 411-423. doi: 10.3934/jimo.2010.6.411

Nonsmooth generalized complementarity as unconstrained optimization

Mohamed Aly Tawhid 1,

1. 

Department of Mathematics and Statistics, Thompson Rivers University, 900 McGill Road, PO Box 3010, Kamloops, BC V2C 5N3, Canada

Received  March 2009 Revised  January 2010 Published  March 2010

We consider generalized complementarity problem GCP$(f,g)$ when the underlying functions $f$ and $g$ are $H$-differentiable. We describe $H$-differentials of some GCP functions and their merit functions. We give some conditions on the $H$-differentials of the given functions under which minimizing a merit function corresponding to such functions leads to a solution of the generalized complementarity problem. Further, we give some conditions on the functions $f$ and $g$ to get a solution of GCP$(f,g)$ by introducing the concepts of relative monotonicity and P0-property and their variants. Our results further give a unified/generalization treatment of such results for the nonlinear complementarity problem when the underlying function is $C^1$ , semismooth, and locally Lipschitzian.
Keywords: Merit function, GCP function, semismooth-functions, locally Lipschitzian, generalized Jacobian, $H$-differentiability., generalized complementarity problem.
Mathematics Subject Classification: Primary: 65K05, 90C33; Secondary: 49J5.
Citation: Mohamed Aly Tawhid. Nonsmooth generalized complementarity as unconstrained optimization. Journal of Industrial & Management Optimization, 2010, 6 (2) : 411-423. doi: 10.3934/jimo.2010.6.411
[1]

Yu-Lin Chang, Jein-Shan Chen, Jia Wu. Proximal point algorithm for nonlinear complementarity problem based on the generalized Fischer-Burmeister merit function. Journal of Industrial & Management Optimization, 2013, 9 (1) : 153-169. doi: 10.3934/jimo.2013.9.153
[2]

Shingo Takeuchi. The basis property of generalized Jacobian elliptic functions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2675-2692. doi: 10.3934/cpaa.2014.13.2675
[3]

Liuyang Yuan, Zhongping Wan, Jingjing Zhang, Bin Sun. A filled function method for solving nonlinear complementarity problem. Journal of Industrial & Management Optimization, 2009, 5 (4) : 911-928. doi: 10.3934/jimo.2009.5.911
[4]

Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091
[5]

Isabelle Déchène. On the security of generalized Jacobian cryptosystems. Advances in Mathematics of Communications, 2007, 1 (4) : 413-426. doi: 10.3934/amc.2007.1.413
[6]

Seung Jun Chang, Jae Gil Choi. Generalized transforms and generalized convolution products associated with Gaussian paths on function space. Communications on Pure & Applied Analysis, 2020, 19 (1) : 371-389. doi: 10.3934/cpaa.2020019
[7]

Boris Muha. A note on the Trace Theorem for domains which are locally subgraph of a Hölder continuous function. Networks & Heterogeneous Media, 2014, 9 (1) : 191-196. doi: 10.3934/nhm.2014.9.191
[8]

Fengming Ma, Yiju Wang, Hongge Zhao. A potential reduction method for the generalized linear complementarity problem over a polyhedral cone. Journal of Industrial & Management Optimization, 2010, 6 (1) : 259-267. doi: 10.3934/jimo.2010.6.259
[9]

Kanji Inui, Hikaru Okada, Hiroki Sumi. The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 753-766. doi: 10.3934/dcds.2020060
[10]

Zsolt Páles, Vera Zeidan. $V$-Jacobian and $V$-co-Jacobian for Lipschitzian maps. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 623-646. doi: 10.3934/dcds.2011.29.623
[11]

Miaohua Jiang. Derivative formula of the potential function for generalized SRB measures of hyperbolic systems of codimension one. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 967-983. doi: 10.3934/dcds.2015.35.967
[12]

Andrey Kochergin. A Besicovitch cylindrical transformation with Hölder function. Electronic Research Announcements, 2015, 22: 87-91. doi: 10.3934/era.2015.22.87
[13]

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
[14]

Anurag Jayswal, Ashish Kumar Prasad, Izhar Ahmad. On minimax fractional programming problems involving generalized $(H_p,r)$-invex functions. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1001-1018. doi: 10.3934/jimo.2014.10.1001
[15]

Bai-Ni Guo, Feng Qi. Properties and applications of a function involving exponential functions. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1231-1249. doi: 10.3934/cpaa.2009.8.1231
[16]

Łukasz Struski, Jacek Tabor. Expansivity implies existence of Hölder continuous Lyapunov function. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3575-3589. doi: 10.3934/dcdsb.2017180
[17]

Kyoungsun Kim, Gen Nakamura, Mourad Sini. The Green function of the interior transmission problem and its applications. Inverse Problems & Imaging, 2012, 6 (3) : 487-521. doi: 10.3934/ipi.2012.6.487
[18]

Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791
[19]

Jianling Li, Chunting Lu, Youfang Zeng. A smooth QP-free algorithm without a penalty function or a filter for mathematical programs with complementarity constraints. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 115-126. doi: 10.3934/naco.2015.5.115
[20]

Hongming Yang, C. Y. Chung, Xiaojiao Tong, Pingping Bing. Research on dynamic equilibrium of power market with complex network constraints based on nonlinear complementarity function. Journal of Industrial & Management Optimization, 2008, 4 (3) : 617-630. doi: 10.3934/jimo.2008.4.617

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences