Skip to main content
SpringerLink
Log in

Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90, 31–43 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Program. 63, 123–145 (1994)

    MathSciNet  MATH  Google Scholar 

  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:10.1155/2010/869684

    MathSciNet  Google Scholar 

  7. Daniele, P., Giannessi, F., Maugeri, A.: Equilibrium problems and variational models. Kluwer (2003)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Facchinei, F., Pang, J. S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2002)

    MATH  Google Scholar 

  10. Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Math, vol. 28. Cambridge University Press, Cambridge (1990)

  11. Halpern, B.: Fixed points of nonexpanding maps. Bull. Amer. Math. Soc. 73, 957–961 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mann, W.R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506–510 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mastroeni, G.: On auxiliary principle for equilibrium problems. Publ. Dipart. Math. Univ. Pisa 3, 1244–1258 (2000)

    Google Scholar 

  15. Muu, L.D., Oettli, W.: Convergence of an adative penalty scheme for finding constrained equilibria. Nonlinear Anal. TMA 18(12), 1159–1166 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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 583082

    Article  MathSciNet  MATH  Google Scholar 

  18. Quoc, T.D., Muu, L.D., Hien, N.V.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)

    Article  MathSciNet  Google Scholar 

  19. Rockafellar R. T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 5, 877–898 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  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) http://www.springerplus.com/content/1/1/8

  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)

    MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  25. Thakur, B.S., Postolache, M.: Existence and approximation of solutions for generalized extended nonlinear variational inequalities. J. Inequal. Appl. 2013 (2013). Art. No. 590

  26. Yao, Postolache, M: Iterative methods for pseudomonotone variational inequalities and fixed point problems. J. Optim. Theory Appl. 155(1), 273–287 (2012)

  27. Yao, Y., Postolache, M., Liou, Y.C.: Variant extragradient-type method for monotone variational inequalities. Fixed Point Theory Appl. 2013 (2013). Art. No. 185

  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. 211

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

    Dang Van Hieu & Pham Ky Anh

  2. Institute of Mathematics, VAST, Hanoi, 18 Hoang Quoc Viet, Hanoi, Vietnam

    Le Dung Muu

Authors
  1. Dang Van Hieu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Le Dung Muu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pham Ky Anh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pham Ky Anh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Van Hieu, D., Muu, L.D. & Anh, P.K. Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings. Numer Algor 73, 197–217 (2016). https://doi.org/10.1007/s11075-015-0092-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-015-0092-5

Keywords

Search

Navigation