Skip to main content
SpringerLink
Log in

A Smoothing Newton Method for Ball Constrained Variational Inequalities with Applications

  • Conference paper
Topics in Numerical Analysis

Part of the book series: Computing Supplementa ((COMPUTING,volume 15))

Abstract

In this paper we present a smoothing Newton method for solving ball constrained variational inequalities where we only require the function value of the underlying function and its derivative value in the feasible region. Global and superlinear convergence of this smoothing Newton method are established. Preliminary numerical results are reported.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Billups, S. C, Dirkse, S. P., Ferris, M. C: A comparison of algorithms for large-scale mixed complementarity problems. Comput. Optim. Appl. 7, 3–25 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  2. Burke, J. V., Xu, S.: A non-interior predictor-corrector path following algorithm for the monotone linear complementarity problem. Math. Programm. 87, 113–130 (2000).

    MathSciNet  MATH  Google Scholar 

  3. Chen, B., Chen, X.: A global and local superlinear continuation-smoothing method for P0 +R0 and monotone NCP. SIAM J. Optim. 9, 624–645 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, C, Mangasarian, O. L.: A class of smoothing functions for nonlinear and mixed complementarity problems. Comput. Optim. Appl.Comput. Optim. Appl. 5, 97–138 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, X., Qi, L., Sun, D.: Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities. Math. Comput. 67, 519–540 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, X., Ye, Y.: On homotopy-smoothing methods for variational inequalities. SIAM J. Control Optim. 37, 589–616 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  7. Clarke F. H.: Optimization and nonsmooth analysis. New York: Wiley, 1983.

    MATH  Google Scholar 

  8. De Luca, T., Facchinei, F., Kanzow, C: A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Programm. 75, 407–439 (1996).

    Article  MATH  Google Scholar 

  9. Facchinei, F., Kanzow, C: A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems. Math. Programm. 76, 493–512 (1997).

    MathSciNet  MATH  Google Scholar 

  10. Fischer, A.: Solution of monotone complementarity problems with locally Lipschitzian functions. Math. Programm. 76, 513–532 (1997).

    MATH  Google Scholar 

  11. Hotta, K., Yoshise, A.: Global convergence of a class of non-interior-point algorithms using Chen-Harker-Kanzow functions for nonlinear complementarity problems. Math. Programm. 86, 105–133 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  12. Jiang, H., Qi, L.: A new nonsmooth equations approach to nonlinear complementarity problems. SIAM J. Control Optim. 55, 178–193 (1997).

    Article  MathSciNet  Google Scholar 

  13. Qi, H.-D.: A regularized smoothing Newton method for box constrained variational inequality problems with P0 functions. SIAM J. Optim. 10, 315–330 (2000).

    Article  MathSciNet  Google Scholar 

  14. Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  15. Qi, L., Sun, D., Zhou, G.: A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities. Math. Programm. 87, 1–35 (2000).

    MathSciNet  MATH  Google Scholar 

  16. Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Programm. 58, 353–367 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  17. Qi, L., Zhou, G.: A smoothing Newton method for minimizing a sum of Euclidean norms. SIAM J. Optim. 11, 389–410 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  18. Robinson, S. M.: Normal maps induced by linear transformation. Math. Oper. Res. 17, 691–71. (1992).

    Article  MathSciNet  MATH  Google Scholar 

  19. Sun, D.: A regularization Newton method for solving nonlinear complementarity problems. Appl. Math. Optim. 40, 315–339 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  20. Xue, G., Ye, Y.: An efficient algorithm for minimizing a sum of Euclidean norms with applications. SIAM J. Optim. 7, 1017–1036 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  21. Xue, G., Ye, Y.: An efficient algorithm for minimizing a sum of P-norms. SIAM J. Optim. 10, 551–579 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  22. Yamashita, N., Fukushima, M.: Modified Newton methods for solving semismooth reformulations of monotone complementarity problems. Math. Programm. 76, 273–284 (1997).

    MathSciNet  Google Scholar 

  23. Zhou, G., Sun, D., Qi, L.: Numerical experiments for a class of squared smoothing Newton methods for box constrained variational inequality problems. In: Reformulation: nonsmooth, piecewise smooth, semismooth and smoothing methods (Fukushima, M., Qi, L., eds.), pp. 421–441. Boston, Kluwer, 1999.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    L. Qi

  2. School of Mathematics, The University of New South Wales, Sydney, 2052, Australia

    G. Zhou

Authors
  1. L. Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Karlsruhe, Karlsruhe, Germany

    Goetz Alefeld

  2. Department of Mathematical Sciences, Shimane University, Matsue, Japan

    Xiaojun Chen

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Wien

About this paper

Cite this paper

Qi, L., Zhou, G. (2001). A Smoothing Newton Method for Ball Constrained Variational Inequalities with Applications. In: Alefeld, G., Chen, X. (eds) Topics in Numerical Analysis. Computing Supplementa, vol 15. Springer, Vienna. https://doi.org/10.1007/978-3-7091-6217-0_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-7091-6217-0_16

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-83673-6

  • Online ISBN: 978-3-7091-6217-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation