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Optimization Letters

, Volume 8, Issue 2, pp 727–738 | Cite as

A hybrid subgradient algorithm for nonexpansive mappings and equilibrium problems

  • P. N. Anh
  • L. D. Muu
Original Paper

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.

Keywords

Equilibrium problems Nonexpansive mappings Pseudomonotone Fixed point 

Mathematics Subject Classification (2010)

65 K10 65 K15 90 C25 90 C33 

Notes

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.

References

  1. 1.
    Anh, P.N.: A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optim. 1, 1–13 (2011)MathSciNetGoogle Scholar
  2. 2.
    Anh, P.N.: A logarithmic quadratic regularization method for solving pseudo-monotone equilibrium problem. ACTA Math. Vietnamica 34, 183–200 (2009)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Anh, P.N., Kim, J.K.: Outer approximation algorithms for pseudomonotone equilibrium problems. Comput. Math. Appl. 61, 2588–2595 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    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)zbMATHMathSciNetGoogle Scholar
  6. 6.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Auslender, A., Teboulle, M., Ben-Tiba, S.: A logaritmic quadratic proximal method for variational inequalities Comput. Optim. Appl. 12, 31–40 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Blum, E., Oettli, W.: From optimization and variational inequality to equilibrium problems. Math. Stud. 63, 127–149 (1994)MathSciNetGoogle Scholar
  9. 9.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekon. Math. Metody 12, 747–756 (1976)zbMATHGoogle Scholar
  10. 10.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)Google Scholar
  11. 11.
    Fukushima, M.: A relaxed projecion method for variational inequalities. Math. Prog. 35, 58–70 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Iusem, A.N., Sosa, W.: Iterative algorithms for equilibrium problems. Optim. 52, 301–316 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Maigé, P.-E.: Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints. European Oper. Res. 205, 501–506 (2010)CrossRefGoogle Scholar
  14. 14.
    Mann, W.R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506–510 (1953)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Mastroeni, G.: Gap functions for equilibrium problems. Glob. Optim. 27, 411–426 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Muu, L.D.: Stability property of a class of variational inequalitites. Optim. 15, 347–351 (1984)zbMATHGoogle Scholar
  17. 17.
    Muu, L.D., Oettli, W.: convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlin. Anal. TMA. 18, 1159–1166 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    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)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Santos, P., Scheimberg, S.: An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 30, 91–107 (2011)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Schu, J.: Weak and strong conergence to fixed points of asymptotically nonexpansive mappings. Bull. Australian Math. Soc. 43, 153–159 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Tada, A., Takahashi, W.: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. Optim. Theory Appl. 133, 359–370 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. Math. Anal. Appl. 331, 506–515 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Tran, Q.D., Muu, L.D., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optim. 57, 749–776 (2008)Google Scholar
  24. 24.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. Math. Anal. Appl. 298, 279–291 (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Scientific FundamentalsPosts and Telecommunications Institute of TechnologyHanoiVietnam
  2. 2.Institute of Mathematics, VASTHanoiVietnam

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