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A Nonmonotone Smoothing Newton Algorithm for Weighted Complementarity Problem

Author

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  • Jingyong Tang

    (Xinyang Normal University)

  • Hongchao Zhang

    (Louisiana State University)

Abstract

The weighted complementarity problem (denoted by WCP) significantly extends the general complementarity problem and can be used for modeling a larger class of problems from science and engineering. In this paper, by introducing a one-parametric class of smoothing functions which includes the weight vector, we propose a smoothing Newton algorithm with nonmonotone line search to solve WCP. We show that any accumulation point of the iterates generated by this algorithm, if exists, is a solution of the considered WCP. Moreover, when the solution set of WCP is nonempty, under assumptions weaker than the Jacobian nonsingularity assumption, we prove that the iteration sequence generated by our algorithm is bounded and converges to one solution of WCP with local superlinear or quadratic convergence rate. Promising numerical results are also reported.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:189:y:2021:i:3:d:10.1007_s10957-021-01839-6
    DOI: 10.1007/s10957-021-01839-6
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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. Zheng-Hai Huang & Tie Ni, 2010. "Smoothing algorithms for complementarity problems over symmetric cones," Computational Optimization and Applications, Springer, vol. 45(3), pages 557-579, April.
    3. Yasushi Narushima & Nobuko Sagara & Hideho Ogasawara, 2011. "A Smoothing Newton Method with Fischer-Burmeister Function for Second-Order Cone Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 149(1), pages 79-101, April.
    4. Li, Yuan-Min & Wei, Deyun, 2015. "A generalized smoothing Newton method for the symmetric cone complementarity problem," Applied Mathematics and Computation, Elsevier, vol. 264(C), pages 335-345.
    5. Nan Lu & Zheng-Hai Huang, 2014. "A Smoothing Newton Algorithm for a Class of Non-monotonic Symmetric Cone Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 161(2), pages 446-464, May.
    6. Defeng Sun & Jie Sun, 2008. "Löwner's Operator and Spectral Functions in Euclidean Jordan Algebras," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 421-445, May.
    7. Xiao-Hong Liu & Zheng-Hai Huang, 2009. "A smoothing Newton algorithm based on a one-parametric class of smoothing functions for linear programming over symmetric cones," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 70(2), pages 385-404, October.
    8. 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, 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.
    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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