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A derivative-free line search technique for Broyden-like method with applications to NCP, wLCP and SI

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  • Jingyong Tang

    (Xinyang Normal University)

  • Jinchuan Zhou

    (Shandong University of Technology)

  • Zhongfeng Sun

    (Shandong University of Technology)

Abstract

We propose a new derivative-free line search technique which contains the classical Li-Fukushima derivative-free line search [Optim. Methods Softw. 13 (3), 181–201, 2000] as a special case. The new line search can enable us to choose a larger step size at each iteration and reduce the number of function evaluations at each step. Based on the new line search, we prove that Broyden-like method for solving the nonlinear equation is globally and locally superlinearly convergent under appropriate assumptions. Moreover, we present some nonlinear equations arising from nonlinear complementarity problems (NCP), weighted linear complementarity problems (wLCP) and system of inequalities (SI). Numerical results show that Broyden-like method based on the new line search has better numerical performance than that based on Li-Fukushima line search.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:annopr:v:321:y:2023:i:1:d:10.1007_s10479-022-04796-z
    DOI: 10.1007/s10479-022-04796-z
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    References listed on IDEAS

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    1. M. Seetharama Gowda, 2019. "Weighted LCPs and interior point systems for copositive linear transformations on Euclidean Jordan algebras," Journal of Global Optimization, Springer, vol. 74(2), pages 285-295, June.
    2. 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.
    3. L. Grippo & F. Rinaldi, 2015. "A class of derivative-free nonmonotone optimization algorithms employing coordinate rotations and gradient approximations," Computational Optimization and Applications, Springer, vol. 60(1), pages 1-33, January.
    4. Jingyong Tang & Jinchuan Zhou, 2021. "A smoothing quasi-Newton method for solving general second-order cone complementarity problems," Journal of Global Optimization, Springer, vol. 80(2), pages 415-438, June.
    5. 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.
    6. Xiaoni Chi & M. Seetharama Gowda & Jiyuan Tao, 2019. "The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra," Journal of Global Optimization, Springer, vol. 73(1), pages 153-169, January.
    7. Bilian Chen & Changfeng Ma, 2011. "A new smoothing Broyden-like method for solving nonlinear complementarity problem with a P 0 -function," Journal of Global Optimization, Springer, vol. 51(3), pages 473-495, November.
    8. 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.
    9. Dong-Hui Li & Masao Fukushima, 2001. "Globally Convergent Broyden-Like Methods for Semismooth Equations and Applications to VIP, NCP and MCP," Annals of Operations Research, Springer, vol. 103(1), pages 71-97, March.
    10. D. H. Li & N. Yamashita & M. Fukushima, 2001. "Nonsmooth Equation Based BFGS Method for Solving KKT Systems in Mathematical Programming," Journal of Optimization Theory and Applications, Springer, vol. 109(1), pages 123-167, April.
    11. Jingyong Tang & Jinchuan Zhou, 2020. "Smoothing inexact Newton method based on a new derivative-free nonmonotone line search for the NCP over circular cones," Annals of Operations Research, Springer, vol. 295(2), pages 787-808, December.
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