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Modified Jacobian smoothing method for nonsmooth complementarity problems

  • Pin-Bo Chen
  • Peng Zhang
  • Xide Zhu
  • Gui-Hua LinEmail author
Article
  • 39 Downloads

Abstract

This paper is devoted to solving a nonsmooth complementarity problem where the mapping is locally Lipschitz continuous but not continuously differentiable everywhere. We reformulate this nonsmooth complementarity problem as a system of nonsmooth equations with the max function and then propose an approximation to the reformulation by simultaneously smoothing the mapping and the max function. Based on the approximation, we present a modified Jacobian smoothing method for the nonsmooth complementarity problem. We show the Jacobian consistency of the function associated with the approximation, under which we establish the global and fast local convergence for the method under suitable assumptions. Finally, to show the effectiveness of the proposed method, we report our numerical experiments on some examples based on MCPLIB/GAMSLIB libraries or network Nash–Cournot game is proposed.

Keywords

Nonsmooth complementarity problem Jacobian consistency Jacobian smoothing method Convergence Network Nash–Cournot game 

Mathematics Subject Classification

90C30 90C33 90C56 

Notes

Acknowledgements

This research was supported in part by National Natural Science Foundation of China (Nos. 11671250, 11901380, 11431004, 71831008) and Humanity and Social Science Foundation of Ministry of Education of China (No. 15YJA630034). The authors are grateful to two anonymous referees for their helpful comments and suggestions, which have led to much improvement of the paper.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Pin-Bo Chen
    • 1
  • Peng Zhang
    • 2
  • Xide Zhu
    • 1
  • Gui-Hua Lin
    • 1
    Email author
  1. 1.School of ManagementShanghai UniversityShanghaiChina
  2. 2.School of Economics and ManagementChongqing University of Posts and TelecommunicationsChongqingChina

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