IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v66y2017i3d10.1007_s10589-016-9881-6.html
   My bibliography  Save this article

A nonmonotone Jacobian smoothing inexact Newton method for NCP

Author

Listed:
  • Sanja Rapajić

    (University of Novi Sad)

  • Zoltan Papp

    (University of Novi Sad)

Abstract

In this paper we propose Jacobian smoothing inexact Newton method for nonlinear complementarity problems (NCP) with derivative-free nonmonotone line search. This nonmonotone line search technique ensures globalization and is a combination of Grippo-Lampariello-Lucidi (GLL) and Li-Fukushima (LF) strategies, with the aim to take into account their advantages. The method is based on very well known Fischer-Burmeister reformulation of NCP and its smoothing Kanzow’s approximation. The mixed Newton equation, which combines the semismooth function with the Jacobian of its smooth operator, is solved approximately in every iteration, so the method belongs to the class of Jacobian smoothing inexact Newton methods. The inexact search direction is not in general a descent direction and this is the reason why nonmonotone scheme is used for globalization. Global convergence and local superlinear convergence of method are proved. Numerical performances are also analyzed and point out that high level of nonmonotonicity of this line search rule enables robust and efficient method.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:66:y:2017:i:3:d:10.1007_s10589-016-9881-6
    DOI: 10.1007/s10589-016-9881-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-016-9881-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10589-016-9881-6?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. 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.
    4. Liqun Qi, 1993. "Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 227-244, February.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. 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.
    3. Pin-Bo Chen & Gui-Hua Lin & Xide Zhu & Fusheng Bai, 2021. "Smoothing Newton method for nonsmooth second-order cone complementarity problems with application to electric power markets," Journal of Global Optimization, Springer, vol. 80(3), pages 635-659, July.
    4. 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.
    5. Dong-Hui Li & Liqun Qi & Judy Tam & Soon-Yi Wu, 2004. "A Smoothing Newton Method for Semi-Infinite Programming," Journal of Global Optimization, Springer, vol. 30(2), pages 169-194, November.
    6. John Duggan & Tasos Kalandrakis, 2011. "A Newton collocation method for solving dynamic bargaining games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 36(3), pages 611-650, April.
    7. Liang Chen & Anping Liao, 2020. "On the Convergence Properties of a Second-Order Augmented Lagrangian Method for Nonlinear Programming Problems with Inequality Constraints," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 248-265, October.
    8. H. Xu & B. M. Glover, 1997. "New Version of the Newton Method for Nonsmooth Equations," Journal of Optimization Theory and Applications, Springer, vol. 93(2), pages 395-415, May.
    9. Ralf Münnich & Ekkehard Sachs & Matthias Wagner, 2012. "Calibration of estimator-weights via semismooth Newton method," Journal of Global Optimization, Springer, vol. 52(3), pages 471-485, March.
    10. Y. Gao, 2006. "Newton Methods for Quasidifferentiable Equations and Their Convergence," Journal of Optimization Theory and Applications, Springer, vol. 131(3), pages 417-428, December.
    11. J. Han & D. Sun, 1997. "Newton and Quasi-Newton Methods for Normal Maps with Polyhedral Sets," Journal of Optimization Theory and Applications, Springer, vol. 94(3), pages 659-676, September.
    12. G. L. Zhou & L. Caccetta, 2008. "Feasible Semismooth Newton Method for a Class of Stochastic Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 139(2), pages 379-392, November.
    13. M. A. Tawhid & J. L. Goffin, 2008. "On Minimizing Some Merit Functions for Nonlinear Complementarity Problems under H-Differentiability," Journal of Optimization Theory and Applications, Springer, vol. 139(1), pages 127-140, October.
    14. C. Kanzow & H. Qi & L. Qi, 2003. "On the Minimum Norm Solution of Linear Programs," Journal of Optimization Theory and Applications, Springer, vol. 116(2), pages 333-345, February.
    15. 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.
    16. Kenji Ueda & Nobuo Yamashita, 2012. "Global Complexity Bound Analysis of the Levenberg–Marquardt Method for Nonsmooth Equations and Its Application to the Nonlinear Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 152(2), pages 450-467, February.
    17. Francisco Facchinei & Christian Kanzow, 2010. "Generalized Nash Equilibrium Problems," Annals of Operations Research, Springer, vol. 175(1), pages 177-211, March.
    18. Zhengyong Zhou & Yunchan Peng, 2019. "The locally Chen–Harker–Kanzow–Smale smoothing functions for mixed complementarity problems," Journal of Global Optimization, Springer, vol. 74(1), pages 169-193, May.
    19. Y. D. Chen & Y. Gao & Y.-J. Liu, 2010. "An Inexact SQP Newton Method for Convex SC1 Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 146(1), pages 33-49, July.
    20. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:coopap:v:66:y:2017:i:3:d:10.1007_s10589-016-9881-6. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.