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An Accelerated Smoothing Newton Method with Cubic Convergence for Weighted Complementarity Problems

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Abstract

Smoothing Newton methods, which usually inherit local quadratic convergence rate, have been successfully applied to solve various mathematical programming problems. In this paper, we propose an accelerated smoothing Newton method (ASNM) for solving the weighted complementarity problem (wCP) by reformulating it as a system of nonlinear equations using a smoothing function. In spirit, when the iterates are close to the solution set of the nonlinear system, an additional approximate Newton step is computed by solving one of two possible linear systems formed by using previously calculated Jacobian information. When a Lipschitz continuous condition holds on the gradient of the smoothing function at two checking points, this additional approximate Newton step can be obtained with a much reduced computational cost. Hence, ASNM enjoys local cubic convergence rate but with computational cost only comparable to standard Newton’s method at most iterations. Furthermore, a second-order nonmonotone line search is designed in ASNM to ensure global convergence. Our numerical experiments verify the local cubic convergence rate of ASNM and show that the acceleration techniques employed in ASNM can significantly improve the computational efficiency compared with some well-known benchmark smoothing Newton method.

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Notes

  1. [39, Lemma 2.6]: Let \(a,b\in {\textbf{V}}\) with \(a\succ _{{\textbf{K}}}0, b\succ _{{\textbf{K}}}0\) and \(a\circ b\succ _{{\textbf{K}}}0.\) Then for all \(u,v\in {\textbf{V}}\) satisfying \(\langle u,v\rangle \ge 0\) and \(a\circ u+b\circ v=0\), we have \(u=v=0\).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, China

    Jingyong Tang

  2. School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255049, China

    Jinchuan Zhou

  3. Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, LA, 70803-4918, USA

    Hongchao Zhang

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  1. Jingyong Tang
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  2. Jinchuan Zhou
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  3. Hongchao Zhang
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Correspondence to Hongchao Zhang.

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Communicated by Dr. Qianchuan Zhao.

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This work was supported by the National Natural Science Foundation of China (11771255), the Shandong Province Natural Science Foundation (ZR2021MA066, 2019KJI013), the Henan Province Natural Science Foundation (222300420520), the Key Scientific Research Projects of Higher Education of Henan Province (22A110020) and the USA National Science Foundation (1819161, 2110722).

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Tang, J., Zhou, J. & Zhang, H. An Accelerated Smoothing Newton Method with Cubic Convergence for Weighted Complementarity Problems. J Optim Theory Appl 196, 641–665 (2023). https://doi.org/10.1007/s10957-022-02152-6

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