Skip to main content
SpringerLink
Log in

Convergence of Hybrid Steepest–Descent Methods for Generalized Variational Inequalities

  • ORIGINAL ARTICLES
  • Published:
Acta Mathematica Sinica Aims and scope Submit manuscript

Abstract

In this paper, we consider the generalized variational inequality GVI(F, g,C), where F and g are mappings from a Hilbert space into itself and C is the fixed point set of a nonexpansive mapping. We propose two iterative algorithms to find approximate solutions of the GVI(F, g,C). Strong convergence results are established and applications to constrained generalized pseudo–inverse are included.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Pang, J. S., Yao, J. C.: On a generalization of a normal map and equations. SIAM J. Control Optim., 33, 168–184 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  2. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980

  3. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems, Springer, New York, 1984

  4. Jaillet, P., Lamberton, D., Lapeyre, B.: Variational inequalities and the pricing of American options. Acta Appl. Math., 21, 263–289 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  5. Oden, J. T.: Qualitative Methods on Nonlinear Mechanics, Prentice–Hall, Englewood Cliffs, New Jersey, 1986

  6. Yao, J. C.: Variational inequalities with generalized monotone operators. Math. Oper. Res., 49, 691–705 (1994)

    Article  Google Scholar 

  7. Konnov, I.: Combined Relaxation Methods for Variational Inequalities, Springer, Berlin, 200

  8. Zeng, L. C.: Completely generalized strongly nonlinear quasi-complementarity problems in Hilbert spaces. J. Math. Anal. Appl., 193, 706–714 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  9. Zeng, L. C.: Iterative algorithm for .nding approximate solutions to completely generalized strongly nonlinear quasivariational inequalities. J. Math. Anal. Appl., 201, 180–194 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Zeng, L. C.: On a general projection algorithm for variational inequalities. J. Optim. Theory Appl., 97, 229–235 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Yamada, I.: The hybrid steepest-descent method for variational inequality problems over the intersection of the .xed point sets of nonexpansive mappings, Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (edited by D. Butnariu, Y. Censor, and S. Reich), North-Holland, Amsterdam, Holland, 473–504, 2001

  12. Deutsch, F., Yamada, I.: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numer. Funct. Anal. Optim., 19, 33–56 (1998)

    MATH  MathSciNet  Google Scholar 

  13. Lions, P. L.: Approximation de points fixes de contractions. C. R. Hebd. Seanc. Acad. Sci. Paris, 284, 1357–1359 (1977)

    MATH  Google Scholar 

  14. Bauschke, H. H.: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl., 202, 150–159 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. Wittmann, R.: Approximation of .xed points of nonexpansive mappings. Arch. Math., 58, 486–491 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  16. Xu, H. K., KIM, T. H.: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl., 119, 185–201 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Xu, H. K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl., 116, 659–678 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Xu, H. K.: Iterative algorithms for nonlinear operators. J. London Math. Soc., 66(2), 240–256 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Goebel, K., Kirk, W. A.: Topics on Metric Fixed–Point Theory. Cambridge University Press, Cambridge, 1990

  20. Bauschke, H. H., Borwein, J. M.: On projection algorithms for solving convex feasibility problems. SIAM Review, 38, 367–426 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  21. Engl, H. W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems, Kluwer, Dordrecht, Holland, 2000

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai 200234, P. R. China

    Liu Chuan Zeng

  2. Department of Applied Mathematics, National Sun Yat–sen University, Kaohsiung, Taiwan 804

    N. C. Wong & J. C. Yao

Authors
  1. Liu Chuan Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. C. Wong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. C. Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liu Chuan Zeng.

Additional information

This project is partially supported both by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, P. R. C., and by the Dawn Program Foundation in Shanghai. Also, this project is partially supported by a grant from the National Science Council of Taiwan, partially supported by Shanghai Leading Academic Discipline Project, Project Number: T0401

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zeng, L.C., Wong, N.C. & Yao, J.C. Convergence of Hybrid Steepest–Descent Methods for Generalized Variational Inequalities. Acta Math Sinica 22, 1–12 (2006). https://doi.org/10.1007/s10114-005-0608-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-005-0608-3

Keywords

MR (2000) Subject Classification

Search

Navigation