Skip to main content
SpringerLink
Log in

The finite intersection property for equilibrium problems

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

The “finite intersection property” for bifunctions is introduced and its relationship with generalized monotonicity properties is studied. Some characterizations are considered involving the Minty equilibrium problem. Also, some results concerning existence of equilibria and quasi-equilibria are established recovering several results in the literature. Furthermore, we give an existence result for generalized Nash equilibrium problems and variational inequality problems.

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

Similar content being viewed by others

Notes

  1. As in the book Functional Analysis by Rudin, a topological vector space includes in its definition that the underlying topology is Hausdorff separated

References

  1. Aussel, D., Cotrina, J.: Quasimonotone quasivariational inequalities: existence results and applications. J. Optim. Theory Appl. 158, 637–652 (2013)

    Article  MathSciNet  Google Scholar 

  2. Aussel, D., Cotrina, J., Iusem, A.: An existence result for quasi-equilibrium problems. J. Convex Anal. 24, 55–66 (2017)

    MathSciNet  MATH  Google Scholar 

  3. Aussel, D., Dutta, J.: Generalized Nash equilibrium problem, variational inequality and quasiconvexity. Oper. Res. Lett. 36(4), 461–464 (2008)

    Article  MathSciNet  Google Scholar 

  4. Aussel, D., Dutta, J., Pandit, T.: About the Links Between Equilibrium Problems and Variational Inequalities, pp. 115–130. Springer, Singapore (2018)

    MATH  Google Scholar 

  5. Aussel, D., Hadjisavvas, N.: On quasimonotone variational inequalities. J. Optim. Theory Appl. 121(2), 445–450 (2004)

    Article  MathSciNet  Google Scholar 

  6. Aussel, D., Hadjisavvas, N.: Adjusted sublevel sets, normal operator, and quasi-convex programming. SIAM J. Optim. 16(2), 358–367 (2005)

    Article  MathSciNet  Google Scholar 

  7. Bianchi, M., Kassay, G., Pini, R.: Existence of equilibria via Ekeland’s principle. J. Math. Anal. 305, 502–512 (2005)

    Article  MathSciNet  Google Scholar 

  8. Bianchi, M., Pini, R.: A note on equilibrium problems with properly quasimonotone bifunctions. J. Glob. Optim. 20, 67–76 (2001)

    Article  MathSciNet  Google Scholar 

  9. Bianchi, M., Pini, R.: Coercivity conditions for equilibrium problems. J. Optim. Theory Appl. 124(1), 79–92 (2005)

    Article  MathSciNet  Google Scholar 

  10. Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90(1), 31–43 (1996)

    Article  MathSciNet  Google Scholar 

  11. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 1–23 (1994)

    MathSciNet  MATH  Google Scholar 

  12. Castellani, M., Giuli, M.: Refinements of existence results for relaxed quasimonotone equilibrium problems. J. Glob. Optim. 57, 1213–1227 (2013)

    Article  MathSciNet  Google Scholar 

  13. Castellani, M., Giuli, M.: An existence result for quasiequilibrium problems in separable Banach spaces. J. Math. Anal. Appl. 425, 85–95 (2015)

    Article  MathSciNet  Google Scholar 

  14. Castellani, M., Giuli, M.: Ekeland’s principle for cyclically monotone equilibrium problems. Nonlinear Anal. RWA 32, 213–228 (2016)

    Article  Google Scholar 

  15. Chaldi, O., Chbani, Z., Riahi, H.: Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities. J. Optim. Theory Appl. 105(2), 299–323 (2000)

    Article  MathSciNet  Google Scholar 

  16. Cotrina, J., García, Y.: Equilibrium problems: existence results and applications. Set-Valued Var. Anal 26, 159–177 (2018)

    Article  MathSciNet  Google Scholar 

  17. Cotrina, J., Zúñiga, J.: Quasi-equilibrium problems with non-self constraint map. J. Glob. Optim. 75, 177–197 (2019)

    Article  MathSciNet  Google Scholar 

  18. Daniilidis, A., Hadjisavvas, N.: On the subdifferentials of quasiconvex and pseudoconvex functions and cyclic monotonicity. J. Math. Anal. Appl. 237, 30–42 (1999)

    Article  MathSciNet  Google Scholar 

  19. Dugundji, J., Granas A.: Fixed point theory. Number v. 1 in Monografie matematyczne. PWN-Polish Scientific Publishers (1982)

  20. Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. 4OR 5(3), 173–210 (2007)

    Article  MathSciNet  Google Scholar 

  21. Fan, K.: A generalization of tychonoff’s fixed point theorem. Mathematische Annalen 142(3), 305–310 (1961)

    Article  MathSciNet  Google Scholar 

  22. Flores-Bazán, F.: Existence theorems for generalized noncoercive equilibrium problems: the quasi-convex case. SIAM J. Optim. 11(3), 675–690 (2001)

    Article  MathSciNet  Google Scholar 

  23. Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. CMS Books in Mathematics. Springer, New York (2003)

    MATH  Google Scholar 

  24. Hadjisavvas, N., Khatibzadeh, H.: Maximal monotonicity of bifunctions. Optimization 59(2), 147–160 (2010)

    Article  MathSciNet  Google Scholar 

  25. Hu, M., Fukushima, M.: Multi-leader-follower games: models, methods and applications. J. Oper. Res. Soc. Jpn. 58(1), 1–23 (2015)

    MathSciNet  MATH  Google Scholar 

  26. Iusem, A.N., Kassay, G., Sosa, W.: On certain conditions for the existence of solutions of equilibrium problems. Math. Program. 116(1), 259–273 (2009)

    Article  MathSciNet  Google Scholar 

  27. Iusem, A.N., Sosa, W.: New existence results for equilibrium problems. Nonlinear Anal. 52(2), 621–635 (2003)

    Article  MathSciNet  Google Scholar 

  28. John, R.: A note on minty variational inequalities and generalized monotonicity. In: Hadjisavvas, N., Martínez-Legaz, J.E., Penot, J.-P. (eds.) Generalized Convexity and Generalized Monotonicity: Proceedings of the 6th International Symposium on Generalized Convexity/Monotonicity, Samos, September 1999, pp. 240–246. Springer, Berlin (2001)

  29. Khanh, P.Q., Quan, N.H.: Versions of the Weierstrass theorem for bifunctions and solution existence in optimization. SIAM J. Optim. 29(2), 1502–1523 (2019)

    Article  MathSciNet  Google Scholar 

  30. Nasri, M., Sosa, W.: Equilibrium problems and generalized Nash games. Optimization 60, 1161–1170 (2011)

    Article  MathSciNet  Google Scholar 

  31. Nessah, R., Tian, G.: Existence of solution of minimax inequalities, equilibria in games and fixed points without convexity and compactness assumptions. J. Optim. Theory Appl. 157, 75–95 (2013)

    Article  MathSciNet  Google Scholar 

  32. Nessah, R., Tian, G.: On the existence of nash equilibrium in discontinuous games. Econ. Theory 61, 515–540 (2016)

    Article  MathSciNet  Google Scholar 

  33. Nikaidô, H., Isoda, K.: Note on noncooperative convex games. Pac. J. Math. 52, 807–815 (1955)

    Article  Google Scholar 

  34. Oettli, W.: A remark on vector-valued equilibria and generalized monotonicity. Acta Math. Vietnam 22, 213–22 (1997)

    MathSciNet  MATH  Google Scholar 

  35. Rosen, J.: Existence and uniqueness of equilibrium points for concave \(n\)-person games. Econometrica 33, 520–534 (1965)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Didier Aussel for his valuable suggestions in Propositions 2, 3, 6 and , Corollary 3 and also Example 4. We also wish to thank the referee for his/her helpful comments and suggestions.

Author information

Authors and Affiliations

  1. Universidad del Pacífico, Lima, Peru

    John Cotrina

  2. Universidad de O’Higgins, Rancagua, Chile

    Anton Svensson

Authors
  1. John Cotrina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anton Svensson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to John Cotrina.

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

Cotrina, J., Svensson, A. The finite intersection property for equilibrium problems. J Glob Optim 79, 941–957 (2021). https://doi.org/10.1007/s10898-020-00961-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-020-00961-5

Keywords

Mathematics Subject Classification

Search

Navigation