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Modified Jacobian smoothing method for nonsmooth complementarity problems

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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.

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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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Authors and Affiliations

  1. School of Management, Shanghai University, Shanghai, 200444, China

    Pin-Bo Chen, Xide Zhu & Gui-Hua Lin

  2. School of Economics and Management, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China

    Peng Zhang

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  1. Pin-Bo Chen
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  2. Peng Zhang
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  3. Xide Zhu
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  4. Gui-Hua Lin
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Correspondence to Gui-Hua Lin.

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Chen, PB., Zhang, P., Zhu, X. et al. Modified Jacobian smoothing method for nonsmooth complementarity problems. Comput Optim Appl 75, 207–235 (2020). https://doi.org/10.1007/s10589-019-00136-3

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