Abstract
We propose a strongly convergent algorithm for finding a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of fixed points of nonexpansive mappings in a real Hilbert space. The proposed algorithm uses only one projection and does not require any Lipschitz condition for the bifunctions.
Similar content being viewed by others
References
Anh, P.N.: A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optim. 1, 1–13 (2011)
Anh, P.N.: A logarithmic quadratic regularization method for solving pseudo-monotone equilibrium problem. ACTA Math. Vietnamica 34, 183–200 (2009)
Anh, P.N., Kim, J.K.: Outer approximation algorithms for pseudomonotone equilibrium problems. Comput. Math. Appl. 61, 2588–2595 (2011)
Anh, P.N., Kim, J.K., Nam, J.M.: Strong convergence of an extragradient method for equilibrium problems and fixed point problems. Korean Math. Soc. 49, 187–200 (2012)
Anh, P.N., Son, D.X.: A new iterative scheme for pseudomonotone equilibrium problems and a finite family of pseudocontractions. Appl. Math. Info. 29, 1179–1191 (2011)
Aoyama, K., Kimura, Y., Takahashi, W., Toyoda, M.: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlin. Anal.TMA. 67, 2350–2360 (2007)
Auslender, A., Teboulle, M., Ben-Tiba, S.: A logaritmic quadratic proximal method for variational inequalities Comput. Optim. Appl. 12, 31–40 (1999)
Blum, E., Oettli, W.: From optimization and variational inequality to equilibrium problems. Math. Stud. 63, 127–149 (1994)
Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekon. Math. Metody 12, 747–756 (1976)
Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)
Fukushima, M.: A relaxed projecion method for variational inequalities. Math. Prog. 35, 58–70 (1986)
Iusem, A.N., Sosa, W.: Iterative algorithms for equilibrium problems. Optim. 52, 301–316 (2003)
Maigé, P.-E.: Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints. European Oper. Res. 205, 501–506 (2010)
Mann, W.R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506–510 (1953)
Mastroeni, G.: Gap functions for equilibrium problems. Glob. Optim. 27, 411–426 (2003)
Muu, L.D.: Stability property of a class of variational inequalitites. Optim. 15, 347–351 (1984)
Muu, L.D., Oettli, W.: convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlin. Anal. TMA. 18, 1159–1166 (1992)
Nguyen, T.T.V., Strodiot, J.J., Nguyen, V.H.: The interior proximal extragradient method for solving equilibrium problems. Glob. Optim. 40, 175–192 (2009)
Santos, P., Scheimberg, S.: An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 30, 91–107 (2011)
Schu, J.: Weak and strong conergence to fixed points of asymptotically nonexpansive mappings. Bull. Australian Math. Soc. 43, 153–159 (1991)
Tada, A., Takahashi, W.: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. Optim. Theory Appl. 133, 359–370 (2007)
Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. Math. Anal. Appl. 331, 506–515 (2007)
Tran, Q.D., Muu, L.D., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optim. 57, 749–776 (2008)
Vuong, P.T., Strodiot, J.J., Nguyen, V.H.: Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems. Optim. Theory Appl. 155, 605–627 (2012)
Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. Math. Anal. Appl. 298, 279–291 (2004)
Acknowledgments
The authors would like to thank the anonymous referees for their helpful and constructive comments and remarks which helped them very much in revising the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported by the Vietnam Institute for Advanced Study in Mathematics.
Rights and permissions
About this article
Cite this article
Anh, P.N., Muu, L.D. A hybrid subgradient algorithm for nonexpansive mappings and equilibrium problems. Optim Lett 8, 727–738 (2014). https://doi.org/10.1007/s11590-013-0612-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-013-0612-y