Abstract
We apply the Banach contraction-mapping fixed-point principle for solving multivalued strongly monotone variational inequalities. Then, we couple this algorithm with the proximal-point method for solving monotone multivalued variational inequalities. We prove the convergence rate of this algorithm and report some computational results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anh, P. N., Muu, L. D., Nguyen, V. H., and Strodiot, J. J., On the Contraction and Nonexpansiveness Properties of the Marginal Mapping in Generalized Variational Inequalities Involving Cocoercive Operators, Generalized Convexity and Generalized Monotonicity, Edited by A. Eberhard, N. Hadjisavvas, P. M. Pardalos, Kluwer Academic Publishers, Dordrecht, Holland, 2004 (to appear).
M. Fukushima (1992) ArticleTitleEquivalent Differentiable Optimization Problems and Descent Methods for Asymmetric Variational Inequality Problems Mathematical Programming 53 99–110
P. Marcotte (1995) ArticleTitleA New Algorithm for Solving Variational Inequalities Mathematical Programming 33 339–351
M. A. Noor (2001) ArticleTitleIterative Schemes for Quasimonotone Mixed Variational Inequalities Optimization 50 29–44
K. Taji M. Fukushima (1996) ArticleTitleA New Merit Function and a Successive Quadratic Programming Algorithm for Variational Inequality Problems SIAM Journal on Optimization 6 704–713
J. H. Wu M. Florian P. Marcotte (1992) ArticleTitleA General Descent Framework for the Monotone Variational Inequality Problem Mathematical Programming 53 99–110
D. Zhu P. Marcotte (1994) ArticleTitleAn Extended Descent Framework for Variational Inequalities Journal of Optimization Theory and Applications 80 349–366
D. Zhu P. Marcotte (1996) ArticleTitleCocoercivity and Its Role in the Convergence of Iterative Schemes for Solving Variational Inequalities SIAM Journal on Optimization 6 714–726
G. Cohen (1988) ArticleTitleAuxiliary Problem Principle Extended to Variational Inequalities Journal of Optimization Theory and Applications 59 325–333
G. Salmon J. J. Strodiot V. H. Nguyen (2004) ArticleTitleA Bundle Method for Solving Variational Inequalities SIAM Journal on Optimization 14 869–893
E. G. Golshtein N. V. Tretyakov (1996) Modified Lagrangians and Monotone Maps in Optimization Wiley New York, NY
I. Konnov (2001) Combined Relaxation Methods for Variational Inequalities Springer New York, NY
B. Martinet (1970) ArticleTitleRégularisation d’Inéquations Variationnelles par Approximations Successives Revue Française d’Automatique et de Recherche Opérationnelle 4 154–159
R. T. Rockafellar (1976) ArticleTitle Monotone Operators and the Proximal-Point Algorithm SIAM Journal on Control and Optimization 14 877–899 Occurrence Handle0358.90053
J. P. Aubin I. Ekeland (1984) Applied Nonlinear Analysis John Wiley and Sons New York, NY
L. D. Muu D. B. Khang (1983) ArticleTitleAsymptotic Regularity and the Strong Convergence of the Proximal-Point Algorithm Acta Mathematica Vietnamica 8 3–11
A. Nagurney (1993) Network Economics: A Variational Inequality Approach Kluwer Academic Publishers Dordrecht, Holland
Author information
Authors and Affiliations
Additional information
This work was completed during the stay of the second author at the Department of Mathematics, University of Namur, Namur, Belgium, 2003.
Rights and permissions
About this article
Cite this article
Anh, P.N., Muu, L.D., Nguyen, V.H. et al. Using the Banach Contraction Principle to Implement the Proximal Point Method for Multivalued Monotone Variational Inequalities. J Optim Theory Appl 124, 285–306 (2005). https://doi.org/10.1007/s10957-004-0926-0
Issue Date:
DOI: https://doi.org/10.1007/s10957-004-0926-0