Weak Convergence of an Iterative Method for Pseudomonotone Variational Inequalities and Fixed-Point Problems
- 182 Downloads
We consider an iterative scheme for finding a common element of the set of solutions of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of N nonexpansive mappings. The proposed iterative method combines two well-known schemes: extragradient and approximate proximal methods. We derive a necessary and sufficient condition for weak convergence of the sequences generated by the proposed scheme.
KeywordsVariational inequalities Nonexpansive mappings Extragradient methods Approximate proximal methods Pseudomonotone mappings Fixed points Weak convergence Opial condition
Unable to display preview. Download preview PDF.
- 1.Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research, vols. I and II. Springer, New York (2003) Google Scholar
- 4.Popov, L.D.: On a one-stage method for solving lexicographic variational inequalities. Izv. Vyssh. Uchebn. Zaved. Mat. 12, 71–81 (1998) Google Scholar
- 6.Yamada, I.: The hybrid steepest-descent method for the variational inequality problem over the intersection of fixed-point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp. 473–504. Kluwer Academic, Dordrecht (2001) Google Scholar
- 9.Ceng, L.C., Yao, J.C.: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwan. J. Math. 10(5), 1293–1303 (2006) Google Scholar
- 14.Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekon. Math. Metody 12, 746–756 (1976); [English translation: Matecon 13, 35–49 (1977)] Google Scholar
- 18.Polyak, B.T.: Introduction to Optimization. Optimization Software Inc., New York (1987) Google Scholar