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



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 {xn} from an arbitrary initial point x0H. The sequence {xn} is shown to converge in norm to the unique solution u of the variational inequality \(\langle F(u^{\ast}), v - u^{\ast}\rangle \geq 0, \quad \forall v \in C.\)

Key Words

Iterative algorithms modified hybrid steepest-descent methods with variable parameters convergence nonexpansive mappings Hilbert spaces 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kinderlehrer, D., and Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, NY, 1980.MATHGoogle Scholar
  2. 2.
    Glowinski, R., Numerical Methods for Nonlinear Variational Problems, Springer, New York, NY, 1984.MATHGoogle Scholar
  3. 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.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Oden, J. T., Qualitative Methods on Nonlinear Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey, 1986.Google Scholar
  5. 5.
    Zeidler, E., Nonlinear Functional Analysis and Its Applications, III: Variational Methods and Applications, Springer, New York, NY, 1985.Google Scholar
  6. 6.
    Yao, J. C., Variational Inequalities with Generalized Monotone Operators, Mathematics of Operations Research, Vol. 19, pp. 691–705, 1994.MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Konnov, I., Combined Relaxation Methods for Variational Inequalities, Springer, Berlin, Germany, 2001.MATHGoogle Scholar
  8. 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.MATHCrossRefMathSciNetGoogle Scholar
  9. 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.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Zeng, L. C., On a General Projection Algorithm for Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 97, pp. 229–235, 1998.MATHCrossRefMathSciNetGoogle Scholar
  11. 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. 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.MATHMathSciNetGoogle Scholar
  13. 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.MATHGoogle Scholar
  14. 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.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Wittmann, R., Approximation of Fixed Points of Nonexpansive Mappings, Archiv der Mathematik, Vol. 58, pp. 486–491, 1992.MATHCrossRefMathSciNetGoogle Scholar
  16. 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.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Xu, H. K., An Iterative Approach to Quadratic Optimization, Journal of Optimization Theory and Applications, Vol. 116, pp. 659–678, 2003.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Goebel, K., and Kirk, W. A., Topics on Metric Fixed-Point Theory, Cambridge University Press, Cambridge, England, 1990.Google Scholar
  19. 19.
    Bauschke, H. H., and Borwein, J. M., On Projection Algorithms for Solving Convex Feasibility Problems, SIAM Review, Vol. 38, pp. 376–426, 1996.CrossRefMathSciNetGoogle Scholar
  20. 20.
    Engl, H. W., Hanke, M., and Neubauer, A., Regularization of Inverse Problems, Kluwer, Dordrecht, Holland, 2000.Google Scholar
  21. 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.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiChina
  2. 2.Department of Applied MathematicsNational Sun Yat-Sen UniversityKaohsiungTaiwan
  3. 3.Department of Applied MathematicsNational Sun Yat-Scn UniversityKaohsiungTaiwan

Personalised recommendations