Abstract
In the present paper, we introduce the concept of η-cocoercivity of a map and develop some iterative schemes for finding the approximate solutions of mixed variational-like inequalities. We use the concept of η-cocoercivity to prove the convergence of the approximate solutions to the exact solution of mixed variational-like inequalities.
Similar content being viewed by others
References
Ding, X. P., Random Mixed Variational-Like Inequalities in Topological Vector Spaces, Journal of the Sichuan Normal University, Vol. 20, pp. 1–13, 1997.
Noor, M. A., Nonconvex Functions and Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 87, pp. 615–630, 1995.
Dien, N. H., Some Remarks on Variational-Like and Quasi-Variational-Like Inequalities, Bulletin of the Australian Mathematical Society, Vol. 46, pp. 335–342, 1992.
Noor, M. A., Variational-Like Inequalities, Optimization, Vol. 30, pp. 323–330, 1994.
Ansari, Q. H., and Yao, J. C., Prevariational Inequalities in Banach Spaces, Optimization: Techniques and Applications, Edited by L. Cacetta et al., Curtin University of Technology, Perth, Australia, Vol. 2, pp. 1165–1172, 1998.
Ansari, Q. H., and Yao, J. C., Nonlinear Variational Inequalities for Pseudomonotone Operators with Applications, Advances in Nonlinear Variational Inequalities, Vol. 3, pp. 61–69, 2000.
Parida, J., Sahoo, M., and Kumar, A., A Variational-Like Inequality Problem, Bulletin of the Australian Mathematical Society, Vol. 39, pp. 225–231, 1989.
Siddiqi, A. H., Khaliq, A., and Ansari, Q. H., On Variational-Like Inequalities, Annales des Sciences Mathématiques du Québec, Vol. 18, pp. 39–48, 1994.
Yang, X. Q., and Chen, G. Y., A Class of Nonconvex Functions and Prevariational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 169, pp. 359–373, 1992.
Cohen, G., Optimization by Decomposition and Coordination: A Unified Approach, IEEE Transactions on Automatic Control, Vol. 23, pp. 222–232, 1978.
Cohen, G., Auxiliary Problem Principle and Decomposition of Optimization Problems, Journal of Optimization Theory and Applications, Vol. 32, pp. 277–305, 1980.
Cohen, G., and Zhu, D. L., Decomposition Coordination Methods in Large-Scale Optimization Problems: The Nondifferentiable Case and the Use of Augmented Lagrangians, Advances in Large-Scale Systems: Theory and Applications, Edited by J. B. Cruz, JAI Press, Greenwich, Connecticut, Vol. 1, pp. 203–266, 1984.
Cohen, G., Auxiliary Problem Principle Extended to Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 49, pp. 325–333, 1988.
Glowinski, R., Lions, J. L., and TrÉmoliÈres, R., Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, Holland, 1981.
Zhu, D. L., and Marcotte, P., Cocoercivity and Its Role in the Convergence of Iterative Schemes for Solving Variational Inequalities, SIAM Journal on Optimization, Vol. 6, pp. 714–726, 1996.
Tseng, P., Further Applications of a Splitting Algorithm to Decomposition in Variational Inequalities and Convex Programming, Mathematical Programming, Vol. 48, pp. 249–263, 1990.
Karamardian, S., The Nonlinear Complementarity Problem with Applications, Part 2, Journal of Optimization Theory and Applications, Vol. 4, pp. 167–181, 1969.
Hanson, M. A., On Sufficiency of the Kuhn–Tucker Conditions, Journal of Mathematical Analysis and Applications, Vol. 80, pp. 545–550, 1981.
KÖthe, G., Topological Vector Spaces, I, Springer Verlag, Berlin, Germany, 1983.
Fan, K., A Generalization of Tychonoff’s Fixed-Point Theorem, Mathematische Annalen, Vol. 142, pp. 305–310, 1961.
Deimling, K., Nonlinear Functional Analysis, Springer Verlag, Berlin, Germany, 1985.
Yao, J. C., Multivalued Variational Inequalities with K-Pseudomonotone Operators, Journal of Optimization Theory and Applications, Vol. 83, pp. 391–403, 1994.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ansari, Q.H., Yao, J.C. Iterative Schemes for Solving Mixed Variational-Like Inequalities. Journal of Optimization Theory and Applications 108, 527–541 (2001). https://doi.org/10.1023/A:1017531323904
Issue Date:
DOI: https://doi.org/10.1023/A:1017531323904