Skip to main content
SpringerLink
Log in

Proximal Point Algorithms for Quasiconvex Pseudomonotone Equilibrium Problems

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We propose a proximal point method for quasiconvex pseudomonotone equilibrium problems. The subproblems of the method are optimization problems whose objective is the sum of a strongly quasiconvex function plus the standard quadratic regularization term for optimization problems. We prove, under suitable additional assumptions, convergence of the generated sequence to a solution of the equilibrium problem, whenever the bifunction is strongly quasiconvex in its second argument, thus extending the validity of the convergence analysis of proximal point methods for equilibrium problems beyond the standard assumption of convexity of the bifunction in the second argument.

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. Bauschke, H.H., Combettes, P.L.: Convex analysis and monotone operators theory in Hilbert spaces. In: CMS Books in Mathematics, 2nd edn. Springer-Verlag, New York (2017)

    Google Scholar 

  2. Beck, A.: First order methods in optimization. In: MOS-SIAM, Series on Optimization. SIAM, Philadelphia (2017)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. Burachik, R., Kassay, G.: On a generalized proximal point method for solving equilibrium problems in Banach spaces. Nonlinear Anal. 75, 6457–6464 (2012)

    MathSciNet  MATH  Google Scholar 

  5. Cambini, A., Martein, L.: Generalized Convexity and Optimization. Springer-Verlag, Berlin-Heidelberg (2009)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Dong, N.T.P., Strodiot, J.J., Van, N.T.T., Nguyen, V.H.: A family of extragradient methods for solving equilibrium problems. J. Ind. Manag. Optim. 11, 619–630 (2015)

    Article  MathSciNet  Google Scholar 

  8. dos Santos Gromicho, J.A.: Quasiconvex Optimization and Location Theory, Kluwer, (1998)

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

    Google Scholar 

  10. Flores-Bazán, F.: Existence theorems for generalized noncoercive equilibrium problems: the quasi-convex case SIAM. J. Optim. 11, 675–690 (2000)

    MathSciNet  MATH  Google Scholar 

  11. Flores-Bazán, F., Flores-Bazán, F., Vera, C.: Maximizing and minimizing quasiconvex functions: related properties, existence and optimality conditions via radial epiderivatives. J. Global Optim. 63, 99–123 (2015)

    Article  MathSciNet  Google Scholar 

  12. Frenk, J.B.G., Schaible, S.: Fractional programming. In: Hadjisavvas, N., et al. (eds.) Handbook of Generalized Convexity and Generalized Monotonicity, pp. 335–386. Springer-Verlag, Boston (2005)

    Chapter  Google Scholar 

  13. Grad, S.-M., Lara, F.: Solving mixed variational inequalities beyond convexity. J. Optim. Theory Appl. 190, 565–580 (2021)

    Article  MathSciNet  Google Scholar 

  14. Hadjisavvas, N.: Convexity, generalized convexity and applications. In: Al-Mezel, S., et al. (eds.) Fixed Point Theory, Variational Analysis and Optimization, pp. 139–169. Taylor & Francis, Boca Raton, Florida (2014)

    Google Scholar 

  15. Hadjisavvas, N., Komlosi, S., Schaible, S.: Handbook of Generalized Convexity and Generalized Monotonicity. Springer-Verlag, Boston (2005)

    Book  Google Scholar 

  16. Hadjisavvas, N., Lara, F., Martínez-Legaz, J.E.: A quasiconvex asymptotic function with applications in optimization. J. Optim. Theory Appl. 180, 170–186 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Iusem, A., Lara, F.: Optimality conditions for vector equilibrium problems with applications. J. Optim. Theory Appl. 180, 187–206 (2019)

    Article  MathSciNet  Google Scholar 

  19. Iusem, A., Lara, F.: Existence results for noncoercive mixed variational inequalities. J. Optim. Theory Appl. 183, 122–138 (2019)

    Article  MathSciNet  Google Scholar 

  20. Iusem, A., Lara, F.: A note on “Existence results for noncoercive mixed variational inequalities in finite dimensional spaces”. J. Optim. Theory Appl. 187, 607–608 (2020)

  21. Iusem, A., Sosa, W.: Iterative algorithms for equilibrium problems. Optimization 52, 301–316 (2003)

    Article  MathSciNet  Google Scholar 

  22. Jeyakumar, V., Oettli, W., Natividad, M.: A solvability theorem for a class of quasiconvex mappings with applications to optimization. J. Math. Anal. Appl. 179, 537–546 (1993)

    Article  MathSciNet  Google Scholar 

  23. Jovanović, M.: A note on strongly convex and quasiconvex functions. Math. Notes 60, 584–585 (1996)

    Article  MathSciNet  Google Scholar 

  24. Kassay, G., Hai, T.N., Vinh, N.T.: Coupling Popov’s algorithm with subgradient extragradient method for solving equilibrium problems. J. Nonlinear Conv. Anal. 19, 959–986 (2018)

  25. Khatibzadeh, H., Mohebbi, V.: Proximal point algorithm for infinite pseudo-monotone bifunctions. Optimization 65, 1629–1639 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. Lara, F.: On strongly quasiconvex functions: existence results and proximal point algorithms. Submitted, (2021)

  28. López, R.: Approximations of equilibrium problems. SIAM J. Optim. 50, 1038–1070 (2012)

    Article  MathSciNet  Google Scholar 

  29. Martinet, B.: Regularisation d’inequations variationelles par approximations successives. Rev. Fr. Inf. Rech. Oper., 154–159 (1970)

  30. Martinet, B.: Determination approchée d’un point fixe d’une application pseudo-contractante. C. R. Acad. Sci. Paris 274, 163–165 (1972)

  31. Mas-Colell, A., Whinston, M.D., Green, J.R.: Microeconomic Theory. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  32. Moudafi, A., Théra, M.: Proximal and Dynamical Approaches to Equilibrium Problems, in Lecture Notes in Econom. and Math. Systems 477, M. Théra and T. Tichatschke (eds.), Springer, Berlin, 187–201, (1999)

  33. Oettli, W.: A remark on vector-valued equilibria and generalized monotonicity. Acta Math. Vietnam. 22, 213–221 (1997)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  35. Poljak, B.T.: Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Sov. Math. 7, 72–75 (1966)

    MATH  Google Scholar 

  36. Quoc, T.D., Muu, L.D., Hien, N.V.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–766 (2008)

    Article  MathSciNet  Google Scholar 

  37. Rockafellar, R.T.: Monotone operators and proximal point algorithms. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MathSciNet  Google Scholar 

  38. Vial, J.P.: Strong convexity of sets and functions. J. Math. Econ. 9, 187–205 (1982)

    Article  MathSciNet  Google Scholar 

  39. Vial, J.P.: Strong and weak convexity of sets and functions. Math. Oper. Res. 8, 231–259 (1983)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research was partially supported by Conicyt–Chile under project Fondecyt Iniciación 11180320 (Lara).

Author information

Authors and Affiliations

  1. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil

    A. Iusem

  2. Departamento de Matemática, Facultad de Ciencias, Universidad de Tarapacá, Arica, Chile

    F. Lara

Authors
  1. A. Iusem
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Lara
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Iusem.

Additional information

Communicated by Qamrul Hasan Ansari.

Honoring Franco Giannessi on his 85th birthday.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Iusem, A., Lara, F. Proximal Point Algorithms for Quasiconvex Pseudomonotone Equilibrium Problems. J Optim Theory Appl 193, 443–461 (2022). https://doi.org/10.1007/s10957-021-01951-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-021-01951-7

Keywords

Search

Navigation