Skip to main content

On fixed point approach to equilibrium problem

Abstract

The equilibrium problem defined by the Nikaidô–Isoda–Fan inequality contains a number of problems such as optimization, variational inequality, Kakutani fixed point, Nash equilibria, and others as special cases. This paper presents a picture for the relationship between the fixed points of the Moreau proximal mapping and the solutions of the equilibrium problem that satisfies some kinds of monotonicity and Lipschitz-type condition.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Agarwal, R.P., O’Regan, D., Sahu, D.R.: Fixed Point Theory for Lipschitzian-type Mappings with Applications. Springer, New York (2009)

    MATH  Google Scholar 

  2. 2.

    Anh, P.K., Hai, T.N.: Splitting extragradient-like algorithms for strongly pseudomonotone equilibrium problems. Numer. Algor. 76(1), 67–91 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Anh, P.N., Hoai An, L.T., Tao, P.D.: Yosida approximation methods for generalized equilibrium problems. J. Convex Anal. 27(3), 959–977 (2020)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Anh, P.N., Muu, L.D., Nguyen, V.H., Strodiot, J.J.: On the contraction and nonexpansiveness properties of the marginal mapping in generalized variational inequalities involving co-coercive operators, Chapter 5. In: Eberhard, A., Hadjisavvas, N., Luc, D.T. (eds.) Generalized Convexity, Generalized Monotonicity and Applications. Nonconvex Optimization and Its Applications, vol. 77. Springer (2005)

  5. 5.

    Anh, T.V., Muu, L.D.: Quasi-nonexpansive mappings involving pseudomonotone bifunctions on convex sets. J. Convex Anal. 25(4), 1105–1119 (2018)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Nonlinear Programming Techniques for Equilibria. Springer, New York (2019)

    MATH  Book  Google Scholar 

  7. 7.

    Bigi, G., Passacantando, M.: Gap functions and penalization for solving equilibrium problems with nonlinear constraints. Comput. Optim. Appl. 53, 323–346 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Bigi, G., Passacantando, M.: D-gap functions and descent techniques for solving equilibrium problems. J. Glob. Optim. 62, 183–203 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Bigi, G., Passacantando, M.: Gap functions for quasi-equilibria. J. Glob. Optim. 66, 791–810 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

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

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

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

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    MATH  Book  Google Scholar 

  13. 13.

    Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces. Springer, New York (2013)

    MATH  Book  Google Scholar 

  14. 14.

    Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Duc, P.M., Muu, L.D.: A splitting algorithm for a class of bilevel equilibrium problems involving nonexpansive mappings. Optimization 65(10), 1855–1866 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Fan, K.: A minimax inequality and applications. In: Shisha, O. (eds.) Inequalities III. Academic Press, pp. 103–113 (1972)

  17. 17.

    Ghosh, M.K., Debnath, L.: Convergence of Ishikawa iterates of quasi-nonexpansive mappings. J. Math. Anal. Appl. 207(1), 96–103 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Hai, T.N.: Contraction of the proximal mapping and applications to the equilibrium problem. Optimization 66(3), 381–396 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Hieu, D.V., Strodiot, J.J., Muu, L.D.: Strongly convergent algorithms by using new adaptive regularization parameter for equilibrium problems. J. Comput. Appl. Math. 376, 112844 (2020)

  20. 20.

    Hung, P.G., Muu, L.D.: The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions. Nonlinear Anal. Theory Methods Appl. 74(17), 6121–6129 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Ishikawa, S.: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44(1), 147–150 (1974)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Kassay, G., Rǎdulescu, V.: Equilibrium Problems and Applications. Academic Press, London (2018)

    MATH  Google Scholar 

  23. 23.

    Konnov, I.V., Pinyagina, O.V.: Descent method with respect to the gap function for nonsmooth equilibrium problems. Russ. Math. 47(12), 67–73 (2003)

    MATH  Google Scholar 

  24. 24.

    Konnov, I.V., Pinyagina, O.V.: D-gap functions and descent methods for a class of monotone equilibrium problems. Lobachevskii J. Math. 13, 57–65 (2003)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Konnov, I.V., Pinyagina, O.V.: D-gap functions for a class of equilibrium problems in Banach spaces. Comput. Methods Appl. Math. 3(2), 274–286 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Konnov, I.V., Schaible, S., Yao, J.C.: Combined relaxation method for mixed equilibrium problems. J. Optim. Theory Appl. 126, 309–322 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Kuhn, H.W., MacKinnon, J.G.: Sandwich method for finding fixed points. J. Optim. Theory Appl. 17, 189–204 (1975)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Langenberg, N.: Interior proximal methods for equilibrium programming: part II. Optimization 62(12), 1603–1625 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Mastroeni, G.: Gap functions for equilibrium problems. J. Glob. Optim. 27(4), 411–426 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Moore, C.: Iterative approximation of fixed points of demicontractive maps. Scientific report IC/98/214, The Abdus Salam International Centre for Theoretical Physics, Trieste (1998)

  31. 31.

    Moreau, J.J.: Proximité et dualité dans un espace Hilbertien. Bull. Soc. Math. Fr. 93, 273–299 (1965)

    MATH  Article  Google Scholar 

  32. 32.

    Muu, L.D., Le, X.T.: A splitting algorithm for finding fixed points of nonexpansive mappings and solving equilibrium problems. J. Fixed Point Theory Appl. 20, 130 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Muu, L.D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. Theory Methods Appl. 18(12), 1159–1166 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Muu, L.D., Quoc, T.D.: One step from DC optimization to DC mixed variational inequalities. Optimization 59(1), 63–76 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Nikaidô, H., Isoda, K.: Note on noncooperative convex games. Pac. J. Math. 5(5), 807–815 (1955)

    MATH  Article  Google Scholar 

  37. 37.

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

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, New York (1998)

    MATH  Book  Google Scholar 

  39. 39.

    Santos, P., Scheimberg, S.: An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 30(1), 91–107 (2011)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Santos, P.S.M., Scheimberg, S.: An outer approximation algorithm for equilibrium problems in Hilbert spaces. Optim. Methods Softw. 30(2), 379–390 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    Scheimberg, S., Santos, P.S.M.: A relaxed projection method for finite-dimensional equilibrium problems. Optimization 60(8–9), 1193–1208 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  42. 42.

    Tada, A., Takahashi, W.: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133, 359–370 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Tan, K.K., Xu, H.K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178(2), 301–308 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Thuy, L.Q., Hai, T.N.: A projected subgradient algorithm for bilevel equilibrium problems and applications. J. Optim. Theory Appl. 175, 411–431 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Tuy, H., Thoai, N.V., Muu, L.D.: A modification of Scarf’s algorithm allowing restarting. Math. Operationsforsch. Stat. Ser. Optim. 9(3), 357–362 (1978)

  46. 46.

    Vinh, N.T., Muu, L.D.: Inertial extragradient algorithms for solving equilibrium problems. Acta Math. Vietnam. 44, 639–663 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    Yen, L.H., Muu, L.D.: A subgradient method for equilibrium problems involving quasiconvex bifunctions. Oper. Res. Lett. 48(5), 579–583 (2020)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the editor and anonymous reviewers for insightful comments and suggestions, that help improve the presentation of the paper. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.01-2020.06.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Xuan Thanh Le.

Additional information

Publisher's Note

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

This is not an open access paper.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Muu, L.D., Le, X.T. On fixed point approach to equilibrium problem. J. Fixed Point Theory Appl. 23, 50 (2021). https://doi.org/10.1007/s11784-021-00890-0

Download citation

Keywords

  • Monotone equilibria
  • fixed point
  • Moreau proximal mapping

Mathematics Subject Classification

  • 47H05
  • 47H10
  • 90C33