Abstract
In this paper, we present new projection methods for solving multivalued variational inequalities on a given nonlinear convex feasible domain. The first one is an extension of the extragradient method to multivalued variational inequalities under the asymptotic optimality condition, but it must satisfy certain Lipschitz continuity conditions. To avoid this requirement, we propose linesearch procedures commonly used in variational inequalities to obtain an approximation linesearch method for solving multivalued variational inequalities. Next, basing on a family of nonempty closed convex subsets of \(\mathcal R^{n}\) and linesearch techniques, we give inner approximation projection algorithms for solving multivalued variational inequalities and the convergence of the algorithms is established under few assumptions.
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References
Anh, P.N.: An interior proximal method for solving pseudomonotone nonlipschitzian multivalued variational inequalities. Nonl. Anal. Forum 14, 27–42 (2009)
Anh, P.N., Kim, J.K.: A new method for solving monotone generalized variational inequalities. J. Ineq. Appl. 2010 (2010). doi:10.1155/2010/657192. Article ID 657192
Anh, P.N., Kim, J.K.: The interior proximal cutting hyperplane method for multivalued variational inequalities. J. Nonl. Conv. Anal. 11, 491–502 (2010)
Anh, P.N., Kuno, T.: A cutting hyperplane method for generalized monotone nonlipschitzian multivalued variational inequalities. In: Bock, H. G., Phu, H. X., Rannacher, R., Schloder, J. P. (eds.) Modeling, Simulation and Optim.ization of Complex Processes. Springer, Berlin (2012)
Anh, P.N., Muu, L.D.: Contraction mapping fixed point algorithms for multivalued mixed variational inequalities on network. In: Dempe, S., Vyacheslav, K. (eds.) Optim.ization with Multivalued Mappings. Springer (2006)
Anh, P.N., Muu, L.D., Strodiot, J.J.: Generalized projection method for non-Lipschitz multivalued monotone variational inequalities. Acta Math. Vietnam. 34, 67–79 (2009)
Anh, P.N., Muu, L.D., Nguyen, V.H., Strodiot, J.J.: On the contraction and nonexpensiveness properties of the marginal mappings in generalized variational inequalities involving cocoercive operators. In: Eberhard, A., Hadjisavvas, N., Luc, D.T. (eds.) Generalized convexity and monotonicity. Springer (2005)
Anh, P.N., Muu, L.D., Nguyen, V.H., Strodiot, J.J.: Using the Banach contraction principle to implement the proximal point method for multivalued monotone variational inequalities. J. Opt. Theory Appl. 124, 285–306 (2005)
Attouch, H.: Variational Convergence for Functions and Operators. Pitman, London (1984)
Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)
Censor, Y., Gibali, A., Reich, S.: Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optim. (2010). doi:10.1080/02331934.2010.539689
Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementary Problems. Springer, New York (2003)
Fanga, C., He, Y.: A double projection algorithm for multivalued variational inequalities and a unified framework of the method. Appl. Math. Comp. 217, 9543–9551 (2011)
Iiduka, H., Yamada, I.: A subgradient-type method for the equilibrium problem over the fixed point set and its applications. Optim. 58, 251–261 (2009)
Iiduka, H., Yamada, I.: An ergodic algorithm for the power-control games for CDMA data networks. J. Math. Model. Algor. 8, 1–18 (2009)
Iusem, A.N., Sosa, W.: Iterative algorithms for equilibrium problems. Optim. 52, 301–316 (2003)
Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Matecon 12, 747–756 (1976)
Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2000)
Taji, K., Fukushima, M.: A new merit function and a successive quadratic programming algorithm for variational inequality problems. SIAM J. Optim. 6, 704–713 (1996)
Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl 331, 506–515 (2007)
Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999)
Strodiot, J.J., Nguyen, T.T.V., Nguyen, V.H.: A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems. J. Global Optim. (2012). doi:10.1007/s10898-011-9814-y
Yao, Y., Noor, M.A., Liou, Y.C., Kang, S.M.: Iterative algorithms for general multivalued variational inequalities. Abstract Appl. Anal. 2012 (2012). doi:10.1155/2012/768272. Article ID 768272
Zheng, L.: The subgradient double projection method for variational inequalities in a Hilbert space. Fixed Point Theory Appl. 2013 (2013). doi:10.1186/1687-1812-2013-136. Article ID 2013:136
Acknowledgments
We are very grateful to the anonymous referees for their really helpful and constructive comments. The work presented here was completed while the first author was on leave at LITA, University of Lorraine, France. He wishes to thank the Fonds Europeens de Developpement Regional Lorraine for the financial support via the FEDER project ”INNOMAD”. This work was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED).
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Anh, P.N., Hoai An, L.T. Outer-Inner Approximation Projection Methods for Multivalued Variational Inequalities. Acta Math Vietnam 42, 61–79 (2017). https://doi.org/10.1007/s40306-015-0165-5
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DOI: https://doi.org/10.1007/s40306-015-0165-5