Skip to main content
SpringerLink
Log in

Projection extragradient algorithms for solving nonmonotone and non-Lipschitzian equilibrium problems in Hilbert spaces

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

Abstract

We present two projection extragradient algorithms for solving equilibrium problems without monotonicity and Lipschitz-type property in Hilbert spaces. Our strategy consists in embedding a subgradient projection step in the extragradient algorithm and employing an Armijo-linesearch. The strategy guarantees that the sequences generated by the presented algorithms converge weakly and strongly to a solution of the equilibrium problem, respectively. The convergence does not require any monotonicity and Lipschitz-type property of the bifunction but the nonemptyness of the solution set of the associated Minty equilibrium problem. Some numerical experiments illustrate the efficiency of the proposed 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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Student. 63, 127–149 (1994)

    MathSciNet  MATH  Google Scholar 

  2. Fan, K.: A minimax inequality and applications. In: Shisha, O (ed.) Inequality III, pp 103–113. Academic Press, New York (1972)

  3. Iusem, A.N., Sosa, W.: New existence results for equilibrium problems. Nonlinear Anal. TMA 52, 621–635 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bigi, G., Pappalardo, M., Passacantando, M.: Existence and solution methods for equilibria. Eur. J. Oper. Res. 227(1), 1–11 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Tran, D.Q., Dung, M.L., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57(6), 749–776 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Strodiot, J.J., Nguyen, T.T.V., Nguyen, V.H.: A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems. J. Glob. Optim. 56, 373–397 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Muu, L.D., Quoc, T.D.: Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash-Cournot equilibrium model. J. Optim. Theory Appl. 142, 185–204 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lions, J.L., Stampacchia, G.: Variational inequalities. Comm. Pure Appl. Math. 20(3), 493–519 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  9. Mastroeni, G.: On auxiliary principle for equilibrium problems. In: Daniele, P., Giannessi, F., Maugeri, A (eds.) Equilibrium Problems and Variational Models, pp 289–298. Kluwer Academic Publishers, Dordrecht (2003)

  10. Flam, S.D., Antipin, A.S.: Equilibrium programming using proximal-like algorithms. Math. Program. 78, 29–41 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Matekon. 12(4), 747–756 (1976)

    MathSciNet  MATH  Google Scholar 

  12. Dinh, B.V., Kim, D.S.: Projection algorithms for solving nonmonotone equilibrium problems in Hilbert space. J. Comput. Appl. Math. 302, 106–117 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ye, M., He, Y.: A double projection method for solving variational inequalities without monotonicity. Comput. Optim. Appl. 60, 141–150 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nguyen, T.T.V., Strodiot, J.J., Nguyen, V.H.: The interior proximal extragradient method for solving equilibrium problems. J. Glob. Optim. 44, 175–192 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Strodiot, J.J., Vuong, P.T., Nguyen, T.T.V.: A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces. J. Glob. Optim. 64, 159–178 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Quoc, T.D., Anh, P.N., Muu, L.D.: Dual extragradient algorithms extended to equilibrium problems. J. Glob. Optim. 52, 139–159 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Iyiola, O.S., Ogbuisi, F.U., Shehu, Y.: An inertial type iterative method with Armijo linesearch for nonomonotone equilibrium problems. Calcolo. 55(4), 1–22 (2018)

    Article  MATH  Google Scholar 

  18. Dinh, B.V., Muu, L.D.: A projection algorithm for solving pseudomonotone equilibrium problems and it’s application to a class of bilevel equilibria. Optimization 64, 559–575 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Vuong, P.T., Strodiot, J.J., Nguyen, V.H.: Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems. J. Optim. Theory Appl. 155, 605–627 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nguyen, T.T.V., Strodiot, J.J., Nguyen, V.H.: Hybrid methods for solving simultaneously an equilibrium problem and countably many fixed point problems in a Hilbert space. J. Optim. Theory Appl. 160, 809–831 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Anh, P.N., An, L.T.H.: The subgradient extragradient method extended to equilibrium problems. Optimization 64(2), 225–248 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality. SIAM J. Control Optim. 37, 765–776 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Iusem, A.N., Svaiter, B.F.: A variant of Korpelevich’s method for variational inequalities with a new search strategy. Optimization 42, 309–321 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  24. Takahashi, W., Takeuchi, Y., Kubota, R.: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 341, 276–286 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Censor, Y., Gibali, A., Reich, S.: Strong convergence of subgradient extragradient methods for variational inequality problem in Hilbert space. Optim. Methods Softw. 26, 827–845 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nakajo, K., Shimoji, K., Takahashi, W.: Strong convergence to common fixed points of families of nonexpansive mappings in Banach spaces. J. Nonlinear Convex Anal. 8, 11–34 (2007)

    MathSciNet  MATH  Google Scholar 

  27. Konnov, I.V., Dyabilkin, D.A.: Non-monotone equilibrium problems: Coercivity conditions and weak regularization. J. Glob. Optim. 49, 575–587 (2011)

    Article  MATH  Google Scholar 

  28. Konnov, I.V.: Application of the proximal point method to nonmonotone equilibrium problems. J. Optim. Theory Appl. 119(2), 317–333 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)

    Book  MATH  Google Scholar 

  30. Bigi, G., Passacantando, M.: Auxiliary problem principles for equilibria. Optimization 66(12), 1955–1972 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yanes, C.M., Xu, H.K.: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal. TMA. 64, 2400–2411 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Vuong, P.T., Strodiot, J.J., Nguyen, V.H.: On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space. Optimization 64 (2), 429–451 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the referees and the editor for their helpful comments and suggestions which have led to the improvement of the early version of this paper.

Funding

This work was partially supported by the National Science Foundation of China (11471230 and 11771067) and the Scientific Research Foundation of the Education Department of Sichuan Province (16ZA0213).

Author information

Authors and Affiliations

  1. College of Mathematics, Sichuan University, Chengdu, Sichuan, People’s Republic of China

    Lanmei Deng & Yaping Fang

  2. College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan, People’s Republic of China

    Rong Hu

Authors
  1. Lanmei Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rong Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yaping Fang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yaping Fang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Deng, L., Hu, R. & Fang, Y. Projection extragradient algorithms for solving nonmonotone and non-Lipschitzian equilibrium problems in Hilbert spaces. Numer Algor 86, 191–221 (2021). https://doi.org/10.1007/s11075-020-00885-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-020-00885-x

Keywords

Search

Navigation