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Globally Convergent Broyden-Like Methods for Semismooth Equations and Applications to VIP, NCP and MCP

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  • Dong-Hui Li
  • Masao Fukushima

Abstract

In this paper, we propose a general smoothing Broyden-like quasi-Newton method for solving a class of nonsmooth equations. Under appropriate conditions, the proposed method converges to a solution of the equation globally and superlinearly. In particular, the proposed method provides the possibility of developing a quasi-Newton method that enjoys superlinear convergence even if strict complementarity fails to hold. We pay particular attention to semismooth equations arising from nonlinear complementarity problems, mixed complementarity problems and variational inequality problems. We show that under certain conditions, the related methods based on the perturbed Fischer–Burmeister function, Chen–Harker–Kanzow–Smale smoothing function and the Gabriel–Moré class of smoothing functions converge globally and superlinearly. Copyright Kluwer Academic Publishers 2001

Suggested Citation

  • 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.
    • Handle: RePEc:spr:annopr:v:103:y:2001:i:1:p:71-97:10.1023/a:1012996232707
    DOI: 10.1023/A:1012996232707
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    Citations

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    Cited by:

    1. Jiao Li & Yu-Fei Yang & Bo Yu, 2012. "A nonmonotone PSB algorithm for solving unconstrained optimization," Computational Optimization and Applications, Springer, vol. 52(1), pages 267-280, May.
    2. 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.
    3. Biao Qu & Changyu Wang & Naihua Xiu, 2017. "Analysis on Newton projection method for the split feasibility problem," Computational Optimization and Applications, Springer, vol. 67(1), pages 175-199, May.
    4. Sanja Rapajić & Zoltan Papp, 2017. "A nonmonotone Jacobian smoothing inexact Newton method for NCP," Computational Optimization and Applications, Springer, vol. 66(3), pages 507-532, April.
    5. Xuebin Wang & Changfeng Ma & Meiyan Li, 2011. "A globally and superlinearly convergent quasi-Newton method for general box constrained variational inequalities without smoothing approximation," Journal of Global Optimization, Springer, vol. 50(4), pages 675-694, August.
    6. Changyu Wang & Qian Liu & Cheng Ma, 2013. "Smoothing SQP algorithm for semismooth equations with box constraints," Computational Optimization and Applications, Springer, vol. 55(2), pages 399-425, June.
    7. 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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