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Modified Jacobian smoothing method for nonsmooth complementarity problems

Author

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  • Pin-Bo Chen

    (Shanghai University)

  • Peng Zhang

    (Chongqing University of Posts and Telecommunications)

  • Xide Zhu

    (Shanghai University)

  • Gui-Hua Lin

    (Shanghai University)

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.

Suggested Citation

  • Pin-Bo Chen & Peng Zhang & Xide Zhu & Gui-Hua Lin, 2020. "Modified Jacobian smoothing method for nonsmooth complementarity problems," Computational Optimization and Applications, Springer, vol. 75(1), pages 207-235, January.
    • Handle: RePEc:spr:coopap:v:75:y:2020:i:1:d:10.1007_s10589-019-00136-3
    DOI: 10.1007/s10589-019-00136-3
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    References listed on IDEAS

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    1. H. Xu, 2001. "Adaptive Smoothing Method, Deterministically Computable Generalized Jacobians, and the Newton Method," Journal of Optimization Theory and Applications, Springer, vol. 109(1), pages 215-224, April.
    2. D. Ralph & H. Xu, 2005. "Implicit Smoothing and Its Application to Optimization with Piecewise Smooth Equality Constraints1," Journal of Optimization Theory and Applications, Springer, vol. 124(3), pages 673-699, March.
    3. Francisco Facchinei & Christian Kanzow, 2010. "Generalized Nash Equilibrium Problems," Annals of Operations Research, Springer, vol. 175(1), pages 177-211, March.
    4. A. Fischer & V. Jeyakumar & D. T. Luc, 2001. "Solution Point Characterizations and Convergence Analysis of a Descent Algorithm for Nonsmooth Continuous Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 110(3), pages 493-513, September.
    5. K. F. Ng & L. L. Tan, 2007. "D-Gap Functions for Nonsmooth Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 133(1), pages 77-97, April.
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    Cited by:

    1. Pin-Bo Chen & Gui-Hua Lin & Xide Zhu & Fusheng Bai, 2021. "Smoothing Newton method for nonsmooth second-order cone complementarity problems with application to electric power markets," Journal of Global Optimization, Springer, vol. 80(3), pages 635-659, July.

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