Numerical Algorithms

, Volume 73, Issue 1, pp 197–217 | Cite as

Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings

  • Dang Van Hieu
  • Le Dung Muu
  • Pham Ky Anh
Original Paper


In this paper we propose and analyze three parallel hybrid extragradient methods for finding a common element of the set of solutions of equilibrium problems involving pseudomonotone bifunctions and the set of fixed points of nonexpansive mappings in a real Hilbert space. Based on parallel computation we can reduce the overall computational effort under widely used conditions on the bifunctions and the nonexpansive mappings. A simple numerical example is given to illustrate the proposed parallel algorithms.


Equilibrium problem Pseudomonotone bifunction Lipschitz-type continuity Nonexpansive mapping Hybrid method Parallel computation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anh, P.N.: A hybrid extragradient method for pseudomonotone equilibrium problems and fixed point problems. Bull. Malays. Math. Sci. Soc. 36(1), 107–116 (2013)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Anh, P.K., Chung, C.V.: Parallel hybrid methods for a finite family of relatively nonexpansive mappings. Numer. Funct. Anal. Optim. 35(6), 649–664 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anh, P.K., Hieu, D.V.: Parallel and sequential hybrid methods for a finite family of asymptotically quasi ϕ-nonexpansive mappings. J. Appl. Math. Comput. 48(1), 241–263 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90, 31–43 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Program. 63, 123–145 (1994)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chang, S.S., Kim, J.K., Wang, X.R.: Modified block iterative algorithm for solving convex feasibility problems in Banach spaces. J. Inequal. Appl. 2010, 869–684 (2010). doi:10.1155/2010/869684MathSciNetGoogle Scholar
  7. 7.
    Daniele, P., Giannessi, F., Maugeri, A.: Equilibrium problems and variational models. Kluwer (2003)Google Scholar
  8. 8.
    Dinh, B.V., Hung, P.G., Muu, L.D.: Bilevel optimization as a regularization approach to pseudomonotone equilibrium problems. Numer. Funct. Anal. Optim. 35 (5), 539–563 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Facchinei, F., Pang, J. S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2002)zbMATHGoogle Scholar
  10. 10.
    Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Math, vol. 28. Cambridge University Press, Cambridge (1990)Google Scholar
  11. 11.
    Halpern, B.: Fixed points of nonexpanding maps. Bull. Amer. Math. Soc. 73, 957–961 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kang, J., Su, Y., Zhang, X.: Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications. Nonlinear Anal. Hybrid Syst. 4, 755–765 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mann, W.R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506–510 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mastroeni, G.: On auxiliary principle for equilibrium problems. Publ. Dipart. Math. Univ. Pisa 3, 1244–1258 (2000)Google Scholar
  15. 15.
    Muu, L.D., Oettli, W.: Convergence of an adative penalty scheme for finding constrained equilibria. Nonlinear Anal. TMA 18(12), 1159–1166 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nadezhkina, N., Takahashi, W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J. Optim. 16, 1230–1241 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Plubtieng, S., Ungchittrakool, K.: Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2008, 19 (2008). Art. ID 583082MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Quoc, T.D., Muu, L.D., Hien, N.V.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Rockafellar R. T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 5, 877–898 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Saewan, S., Kumam, P.: The hybrid block iterative algorithm for solving the system of equilibrium problems and variational inequality problems. Saewan and Kumam Springer Plus 2012 (2012)
  21. 21.
    Su, Y., Li, M., Zhang, H.: New monotone hybrid algorithm for hemi-relatively nonexpansive mappings and maximal monotone operators. Appl. Math. Comput. 217 (12), 5458–5465 (2011)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Tada, A., Takahashi, W.: Strong convergence theorem for an equilibrium problem and a nonexpansive mapping. In: Takahashi, W., Tanaka, T. (eds.) Nonlinear Analysis and Convex Analysis. Yokohama Publishers, Yokohama (2006)Google Scholar
  23. 23.
    Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point in Hilbert space. J. Math. Anal. Appl. 331(1), 506–515 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Tan, K.K., Xu, H.K.: Approximating fixed points of nonexpansive by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301–308 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Thakur, B.S., Postolache, M.: Existence and approximation of solutions for generalized extended nonlinear variational inequalities. J. Inequal. Appl. 2013 (2013). Art. No. 590Google Scholar
  26. 26.
    Yao, Postolache, M: Iterative methods for pseudomonotone variational inequalities and fixed point problems. J. Optim. Theory Appl. 155(1), 273–287 (2012)Google Scholar
  27. 27.
    Yao, Y., Postolache, M., Liou, Y.C.: Variant extragradient-type method for monotone variational inequalities. Fixed Point Theory Appl. 2013 (2013). Art. No. 185Google Scholar
  28. 28.
    Yao, Y., Postolache, M., Liou, Y.C.: Coupling Ishikawa algorithms with hybrid techniques for pseudocontractive mappings. Fixed Point Theory Appl. 2013 (2013). Art. No. 211Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsVietnam National UniversityHanoiVietnam
  2. 2.Institute of Mathematics, VASTHanoiVietnam

Personalised recommendations