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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 531912, 7 pages
An Extension of Subgradient Method for Variational Inequality Problems in Hilbert Space
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
Received 26 October 2012; Revised 30 January 2013; Accepted 1 February 2013
Academic Editor: Guanglu Zhou
Copyright © 2013 Xueyong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- R. W. Cottle, J.-S. Pang, and R. E. Stone, The Linear Complementarity Problem, Academic Press, Boston, Mas, USA, 1992.
- G. M. Korpelevich, “The extragradient method for finding saddle points and other problems,” Matecon, vol. 12, pp. 747–756, 1976.
- M. V. Solodov and B. F. Svaiter, “A new projection method for variational inequality problems,” SIAM Journal on Control and Optimization, vol. 37, no. 3, pp. 765–776, 1999.
- M. V. Solodov, “Stationary points of bound constrained minimization reformulations of complementarity problems,” Journal of Optimization Theory and Applications, vol. 94, no. 2, pp. 449–467, 1997.
- Y. J. Wang, N. H. Xiu, and J. Z. Zhang, “Modified extragradient method for variational inequalities and verification of solution existence,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 167–183, 2003.
- W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428, 2003.
- E. H. Zarantonello, Projections on Convex Sets in Hilbert Space and Spectral Theory, Contributions to Nonlinear Functional Analysis, Academic press, New York, NY, USA, 1971.
- Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.
- R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, pp. 75–88, 1970.
- F. E. Browder, “Fixed-point theorems for noncompact mappings in Hilbert space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 53, pp. 1272–1276, 1965.
- N. Nadezhkina and W. Takahashi, “Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings,” Journal of Optimization Theory and Applications, vol. 128, no. 1, pp. 191–201, 2006.
- B. C. Eaves, “On the basic theorem of complementarity,” Mathematical Programming, vol. 1, no. 1, pp. 68–75, 1971.
- B. S. He and L. Z. Liao, “Improvements of some projection methods for monotone nonlinear variational inequalities,” Journal of Optimization Theory and Applications, vol. 112, no. 1, pp. 111–128, 2002.
- Y. Censor, A. Gibali, and S. Reich, “The subgradient extragradient method for solving variational inequalities in Hilbert space,” Journal of Optimization Theory and Applications, vol. 148, no. 2, pp. 318–335, 2011.
- Y. Censor, A. Gibali, and S. Reich, “Two extensions of Korpelevich’s extragradient method for solving the variational inequality problem in Euclidean space,” Tech. Rep., 2010.
- Y. J. Wang, N. H. Xiu, and C. Y. Wang, “Unified framework of extragradient-type methods for pseudomonotone variational inequalities,” Journal of Optimization Theory and Applications, vol. 111, no. 3, pp. 641–656, 2001.
- J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 153–159, 1991.