IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v113y2002i1d10.1023_a1014861331301.html
   My bibliography  Save this article

Smoothing Functions and Smoothing Newton Method for Complementarity and Variational Inequality Problems

Author

Listed:
  • L. Qi

    (Hong Kong Polytechnic University)

  • D. Sun

    (National University of Singapore)

Abstract

This paper provides for the first time some computable smoothing functions for variational inequality problems with general constraints. This paper proposes also a new version of the smoothing Newton method and establishes its global and superlinear (quadratic) convergence under conditions weaker than those previously used in the literature. These are achieved by introducing a general definition for smoothing functions, which include almost all the existing smoothing functions as special cases.

Suggested Citation

  • L. Qi & D. Sun, 2002. "Smoothing Functions and Smoothing Newton Method for Complementarity and Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 113(1), pages 121-147, April.
    • Handle: RePEc:spr:joptap:v:113:y:2002:i:1:d:10.1023_a:1014861331301
    DOI: 10.1023/A:1014861331301
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/A:1014861331301
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1023/A:1014861331301?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. James V. Burke & Song Xu, 1998. "The Global Linear Convergence of a Noninterior Path-Following Algorithm for Linear Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 23(3), pages 719-734, August.
    2. Stephen M. Robinson, 1992. "Normal Maps Induced by Linear Transformations," Mathematics of Operations Research, INFORMS, vol. 17(3), pages 691-714, August.
    3. M. Seetharama Gowda & Roman Sznajder, 1999. "Weak Univalence and Connectedness of Inverse Images of Continuous Functions," Mathematics of Operations Research, INFORMS, vol. 24(1), pages 255-261, February.
    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)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Yiyin Cao & Yin Chen & Chuangyin Dang, 2024. "A Variant of the Logistic Quantal Response Equilibrium to Select a Perfect Equilibrium," Journal of Optimization Theory and Applications, Springer, vol. 201(3), pages 1026-1062, June.
    2. Louis Caccetta & Biao Qu & Guanglu Zhou, 2011. "A globally and quadratically convergent method for absolute value equations," Computational Optimization and Applications, Springer, vol. 48(1), pages 45-58, January.
    3. Nina Ovcharova & Joachim Gwinner, 2014. "A Study of Regularization Techniques of Nondifferentiable Optimization in View of Application to Hemivariational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 162(3), pages 754-778, September.
    4. Kamil A. Khan & Harry A. J. Watson & Paul I. Barton, 2017. "Differentiable McCormick relaxations," Journal of Global Optimization, Springer, vol. 67(4), pages 687-729, April.
    5. Jinhai Chen & Herschel Rabitz, 2019. "On Lifting Operators and Regularity of Nonsmooth Newton Methods for Optimal Control Problems of Differential Algebraic Equations," Journal of Optimization Theory and Applications, Springer, vol. 180(2), pages 518-535, February.
    6. Chen Ling & Hongxia Yin & Guanglu Zhou, 2011. "A smoothing Newton-type method for solving the L 2 spectral estimation problem with lower and upper bounds," Computational Optimization and Applications, Springer, vol. 50(2), pages 351-378, October.
    7. 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.
    8. 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.
    9. Zhengyong Zhou & Bo Yu, 2014. "A smoothing homotopy method for variational inequality problems on polyhedral convex sets," Journal of Global Optimization, Springer, vol. 58(1), pages 151-168, January.
    10. Xiaona Fan & Qinglun Yan, 2018. "A New Proof for Global Convergence of a Smoothing Homotopy Method for the Nonlinear Complementarity Problem," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 35(04), pages 1-13, August.
    11. C. Zhang & Q. J. Wei, 2009. "Global and Finite Convergence of a Generalized Newton Method for Absolute Value Equations," Journal of Optimization Theory and Applications, Springer, vol. 143(2), pages 391-403, November.

    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. 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.
    2. 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.
    3. 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.
    4. Todd S. Munson & Francisco Facchinei & Michael C. Ferris & Andreas Fischer & Christian Kanzow, 2001. "The Semismooth Algorithm for Large Scale Complementarity Problems," INFORMS Journal on Computing, INFORMS, vol. 13(4), pages 294-311, November.
    5. S. H. Pan & Y. X. Jiang, 2008. "Smoothing Newton Method for Minimizing the Sum of p-Norms," Journal of Optimization Theory and Applications, Springer, vol. 137(2), pages 255-275, May.
    6. Yun-Bin Zhao & Duan Li, 2001. "On a New Homotopy Continuation Trajectory for Nonlinear Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 26(1), pages 119-146, February.
    7. Michael Patriksson & R. Tyrrell Rockafellar, 2002. "A Mathematical Model and Descent Algorithm for Bilevel Traffic Management," Transportation Science, INFORMS, vol. 36(3), pages 271-291, August.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. Baohua Huang & Wen Li, 2023. "A smoothing Newton method based on the modulus equation for a class of weakly nonlinear complementarity problems," Computational Optimization and Applications, Springer, vol. 86(1), pages 345-381, September.
    14. 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.
    15. 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.
    16. 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.
    17. 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.
    18. 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.
    19. Alexander Shapiro, 2005. "Sensitivity Analysis of Parameterized Variational Inequalities," Mathematics of Operations Research, INFORMS, vol. 30(1), pages 109-126, February.
    20. 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.

    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:joptap:v:113:y:2002:i:1:d:10.1023_a:1014861331301. 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.