Skip to main content
SpringerLink
Log in

Convergence Analysis of Modified Hybrid Steepest-Descent Methods with Variable Parameters for Variational Inequalities

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

Assume that F is a nonlinear operator on a real Hilbert space H which is η-strongly monotone and κ-Lipschitzian on a nonempty closed convex subset C of H. Assume also that C is the intersection of the fixed-point sets of a finite number of nonexpansive mappings on H. We construct an iterative algorithm with variable parameters which generates a sequence {x n } from an arbitrary initial point x 0 ∊ H. The sequence {x n } is shown to converge in norm to the unique solution u ∗ of the variational inequality \(\langle F(u^{\ast}), v - u^{\ast}\rangle \geq 0, \quad \forall v \in C.\)

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

    MATH  Google Scholar 

  2. Glowinski, R., Numerical Methods for Nonlinear Variational Problems, Springer, New York, NY, 1984.

    MATH  Google Scholar 

  3. Jaillet, P., Lamberton, D., and Lapeyre, B., Variational Inequalities and the Princing of American Options, Acta Applicandae Mathematicae, Vol. 21, pp. 263–289, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  4. Oden, J. T., Qualitative Methods on Nonlinear Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey, 1986.

    Google Scholar 

  5. Zeidler, E., Nonlinear Functional Analysis and Its Applications, III: Variational Methods and Applications, Springer, New York, NY, 1985.

    Google Scholar 

  6. Yao, J. C., Variational Inequalities with Generalized Monotone Operators, Mathematics of Operations Research, Vol. 19, pp. 691–705, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  7. Konnov, I., Combined Relaxation Methods for Variational Inequalities, Springer, Berlin, Germany, 2001.

    MATH  Google Scholar 

  8. Zeng, L. C., Iterative Algorithm for Finding Approximate Solutions to Completely Generalized Strongly Nonlinear Quasivariational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 201, pp. 180–194, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  9. Zeng, L. C., Completely Generalized Strongly Nonlinear Quasicomplementarity Problems in Hilbert Spaces, Journal of Mathematical Analysis and Applications, Vol. 193, pp. 706–714, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  10. Zeng, L. C., On a General Projection Algorithm for Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 97, pp. 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 Fixed-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, pp. 473–504, 2001.

    Google Scholar 

  12. Deutsch, F., and Yamada. I., Minimizing Certain Convex Functions over the Intersection of the Fixed-Point Sets of Nonexpansive Mappings, Numerical Functional Analysis and Optimization, Vol. 19, pp. 33–56, 1998.

    MATH  MathSciNet  Google Scholar 

  13. Lions, P.L., Approximation de Points Fixes de Contractions, Comptes Rendus de L'Academie des Sciences de Paris, Vol. 284, pp. 1357–1359, 1977.

    MATH  Google Scholar 

  14. Bauschke, H. H., The Approximation of Fixed Points of Compositions of Nonexpansive Mappings in Hilbert Spaces, Journal of Mathematical Analysis and Applications, Vol. 202, pp. 150–159, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  15. Wittmann, R., Approximation of Fixed Points of Nonexpansive Mappings, Archiv der Mathematik, Vol. 58, pp. 486–491, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  16. Xu, H. K., and Kim, T. H., Convergence of Hybrid Steepest-Descent Methods for Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 119, pp. 185–201, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  17. Xu, H. K., An Iterative Approach to Quadratic Optimization, Journal of Optimization Theory and Applications, Vol. 116, pp. 659–678, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  18. Goebel, K., and Kirk, W. A., Topics on Metric Fixed-Point Theory, Cambridge University Press, Cambridge, England, 1990.

    Google Scholar 

  19. Bauschke, H. H., and Borwein, J. M., On Projection Algorithms for Solving Convex Feasibility Problems, SIAM Review, Vol. 38, pp. 376–426, 1996.

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  21. Yamada, I., Ogura, N., and Shirakawa, N., A Numerically Robust Hybrid Steepest Descent Method for Convexly Constrained Generalized Inverse Problems, Inverse Problems, Image Analysis, and Medical Imaging, Contemporary Mathematics, Edited by Z. Nashed and O. Scherzer, Vol. 313, pp. 269–305, 2002.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, China

    L. C. Zeng (Professor)

  2. Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan

    N. C. Wong (Professor)

  3. Department of Applied Mathematics, National Sun Yat-Scn University, Kaohsiung, Taiwan

    J. C. Yao (Professor)

Authors
  1. L. C. 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 J. C. Yao.

Additional information

The authors thank the referees for helpful comments and suggestions

His research was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and the Dawn Program Foundation in Shanghai

His research was partially supported by grant from the National Science Council of Taiwan

His research was partially supported by grant from the National Science Council of Taiwan

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zeng, L.C., Wong, N.C. & Yao, J.C. Convergence Analysis of Modified Hybrid Steepest-Descent Methods with Variable Parameters for Variational Inequalities. J Optim Theory Appl 132, 51–69 (2007). https://doi.org/10.1007/s10957-006-9068-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-006-9068-x

Key Words

Search

Navigation