Abstract
This paper introduces and considers a new system of generalized mixed variational inequalities in a Hilbert space, which includes many new and known systems of variational inequalities and generalized variational inequalities as special cases. By using the two concepts of η-subdifferential and η-proximal mappings of a proper function, the authors try to demonstrate that the system of generalized mixed variational inequalities is equivalence with a fixed point problem. By applying the equivalence, a new and innovative η-proximal point algorithm for finding approximate solutions of the system of generalized mixed variational inequalities will be suggested and analyzed. The authors also study the convergence analysis of the new iterative method under much weaker conditions. The results can be viewed as a refinement and improvement of the previously known results for variational inequalities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. U. Verma, General convergence analysis for two-step projection methods and application to variational problems, Appl. Math. Lett., 2005, 18: 1286–1292.
S. S. Chang, H. W. J. Lee, and C. K. Chan, Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces, Appl. Math. Lett., 2007, 20: 329–334.
Z. Y. Huang and M. A. Noor, An explicit projection method for a system of nonlinear variational inequalities with different (γ, r)-cocoercive mappings, Applied Mathematics and Computation, 2007, 190: 356–361.
M. A. Noor and K. I. Noor, Projection algorithms for solving a system of general variational inequalities, Nonlinear Analysis, 2009, 70: 2700–2706.
X. P. Ding and C. L. Luo, Perturbed Proximal point algorithms for general quasi-variational-like inclusions, J. Comput. Appl. Math., 2000, 113: 153–165.
X. P. Ding, Generalized quasi-variational-like inclusions with nonconvex functionals, Applied Mathematics and Computation, 2001, 122: 267–282.
M. A. Noor, Mixed quasi variational inequalities, Applied Mathematics and Computation, 2003, 146: 553–578.
Abdellah Bnouhachem, M. A. Noor, and T. M. Rassias, Three-steps iterative algorithms for mixed variational inequalities, Applied Mathematics and Computation, 2006, 183: 436–446.
M. A. Noor, Projection-proximal methods for general variational inequalities, J. Math. Anal. Appl., 2006, 318: 53–62.
M. A. Noor, Some developments in general variational inequalities, Appl. Math. Comput., 2004, 152: 199–277.
M. A. Noor, Differentiable nonconvex functions and general variational inequalities, Appl. Math. Comput., 2008, 199: 623–630.
S. S. Chang, Variational Inequality and Complementarity Problem Theory with Applications, Shanghai Scientific and Technology Literature, Shanghai, 1991.
A. Hassouni and A. Moudafi, A perturbed algorithm for variational inclusions, J. Math. Anal. Appl., 1994, 185: 706–712.
M. A. Noor, General variational inequalities, Appl. Math. Lett., 1988, 1: 119–121.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research is supported by the Natural Science Foundation of China under Grant No. 11001287 and the Natural Science Foundation Project of CSTC under Grant No. 2010BB9254.
This paper was recommended for publication by Editor Shouyang WANG.
Rights and permissions
About this article
Cite this article
Wan, B., Zhan, X. A proximal point algorithm for a system of generalized mixed variational inequalities. J Syst Sci Complex 25, 964–972 (2012). https://doi.org/10.1007/s11424-012-0157-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-012-0157-7