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Quadratic convergence analysis of a nonmonotone Levenberg–Marquardt type method for the weighted nonlinear complementarity problem

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  • Jingyong Tang

    (Xinyang Normal University)

  • Jinchuan Zhou

    (Shandong University of Technology)

Abstract

In this paper we consider the weighted nonlinear complementarity problem (denoted by wNCP) which contains a wide class of optimization problems. We introduce a family of new weighted complementarity functions and show that it is continuously differentiable everywhere and has several favorable properties. Based on this function, we reformulate the wNCP as a smooth nonlinear equation and propose a nonmonotone Levenberg–Marquardt type method to solve it. We show that the proposed method is well-defined and it is globally convergent without any additional condition. Moreover, we prove that the whole iteration sequence converges to a solution of the wNCP locally superlinearly or quadratically under the nonsingularity condition. In addition, we establish the local quadratic convergence of the proposed method under the local error bound condition. Some numerical results are also reported.

Suggested Citation

  • Jingyong Tang & Jinchuan Zhou, 2021. "Quadratic convergence analysis of a nonmonotone Levenberg–Marquardt type method for the weighted nonlinear complementarity problem," Computational Optimization and Applications, Springer, vol. 80(1), pages 213-244, September.
    • Handle: RePEc:spr:coopap:v:80:y:2021:i:1:d:10.1007_s10589-021-00300-8
    DOI: 10.1007/s10589-021-00300-8
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    References listed on IDEAS

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    1. Soodabeh Asadi & Zsolt Darvay & Goran Lesaja & Nezam Mahdavi-Amiri & Florian Potra, 2020. "A Full-Newton Step Interior-Point Method for Monotone Weighted Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 864-878, September.
    2. Z.H. Huang & J. Han & Z. Chen, 2003. "Predictor-Corrector Smoothing Newton Method, Based on a New Smoothing Function, for Solving the Nonlinear Complementarity Problem with a P 0 Function," Journal of Optimization Theory and Applications, Springer, vol. 117(1), pages 39-68, April.
    3. Y. Q. Bai & G. Lesaja & C. Roos & G. Q. Wang & M. El Ghami, 2008. "A Class of Large-Update and Small-Update Primal-Dual Interior-Point Algorithms for Linear Optimization," Journal of Optimization Theory and Applications, Springer, vol. 138(3), pages 341-359, September.
    4. Changfeng Ma, 2010. "A new smoothing and regularization Newton method for P 0 -NCP," Journal of Global Optimization, Springer, vol. 48(2), pages 241-261, October.
    5. Jingyong Tang & Hongchao Zhang, 2021. "A Nonmonotone Smoothing Newton Algorithm for Weighted Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 189(3), pages 679-715, June.
    6. Florian A. Potra, 2016. "Sufficient weighted complementarity problems," Computational Optimization and Applications, Springer, vol. 64(2), pages 467-488, June.
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    Cited by:

    1. Jingyong Tang & Jinchuan Zhou & Hongchao Zhang, 2023. "An Accelerated Smoothing Newton Method with Cubic Convergence for Weighted Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 196(2), pages 641-665, February.
    2. Jingyong Tang & Jinchuan Zhou & Zhongfeng Sun, 2023. "A derivative-free line search technique for Broyden-like method with applications to NCP, wLCP and SI," Annals of Operations Research, Springer, vol. 321(1), pages 541-564, February.

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