Skip to main content
SpringerLink
Log in

One-Step iterative method for bilevel equilibrium problem in Hilbert space

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

Abstract

The purpose of this paper is to introduce a simple iterative method for finding a solution of an equilibrium problem whose constraint is the solution set of another monotone equilibrium problem in a Hilbert space. Unlike the multi-step methods, the new method only requires to find one value of the proximal mapping associated with cost bifunctions at the current approximation over each iterative step. The strong convergence of the iterative sequence generated by the method is established by incorporating with a regularization technique. The numerical behavior of our method is also illustrated in comparison with several other methods.

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

Similar content being viewed by others

References

  1. Ait Mansour, M., Elakri, R.-A., Laghdir, M.: Quantitative stability for equilibrium problems under relaxed upper-sign properties: Application to quasiconvex programming. Appl. Anal. Optim. 3, 307–331 (2019)

    MATH  Google Scholar 

  2. Ait Mansour, M., Riahi, H.: Sensitivity analysis for abstract equilibrium problems. J. Math. Anal. Appl. 306, 684–691 (2005)

    Article  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  4. Bigi, G., Passacantando, M.: Descent and penalization techniques for equilibrium problems with nonlinear constraints. J. Optim. Theory Appl. 164, 804–818 (2015)

    Article  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  8. Chbani, Z., Riahi, H.: Weak and strong convergence of prox-penalization and splitting algorithms for bilevel equilibrium problems. Numer. Algebra Control Optim. 3, 353–366 (2013)

    Article  MATH  Google Scholar 

  9. Chbani, Z., Riahi, H.: Weak and strong convergence of proximal penalization and proximal splitting algorithms for two-level hierarchical Ky Fan minimax inequalities. Optim. 64, 1285–1303 (2015)

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  11. Dempe, S.: Foundations of Bilevel Programming. Kluwer Academic Publishers (2002)

  12. Dempe, S.: Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52, 333–359 (2003)

    Article  MATH  Google Scholar 

  13. Dempe, S., Kalashnikov, V., Pérez-Valdés, G.A., Kalashnykova, N.: Bilevel programming problems. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  14. Dinh, B.V., Muu, L.D.: On penalty and gap function methods for bilevel equilibrium problems. J. Appl. Math. 2011, 1–14 (2011). https://doi.org/10.1155/2011/646452

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  16. Hieu, D.V., Cho, Y.J., Xiao, Y.B.: Modified extragradient algorithms for solving equilibrium problems. Optimization. 67, 2003–2029 (2018)

    Article  MATH  Google Scholar 

  17. Hieu, D.V., Quy, P.K., Hong, L.T., Vy, L.V.: Accelerated hybrid methods for solving pseudomonotone equilibrium problems. Adv. Comput. Math 46, 58 (2020)

    Article  MATH  Google Scholar 

  18. Hieu, D.V.: Strong convergence of a new hybrid algorithm for fixed point problems and equilibrium problems. Math. Modell. Anal. 24, 1–19 (2019)

    Article  MATH  Google Scholar 

  19. Hieu, D.V., Strodiot, J.J.: Strong convergence theorems for equilibrium problems and fixed point problems in Banach spaces. J. Fixed Point Theory Appl. 20, 131 (2018)

    Article  MATH  Google Scholar 

  20. 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). https://doi.org/10.1016/j.cam.2020.112844

    Article  MATH  Google Scholar 

  21. Hieu, D.V., Muu, L.D., Quy, P.K., Duong, H.N.: Regularization extragradient methods for equilibrium programming in Hilbert spaces. Optim. (2021). https://doi.org/10.1080/02331934.2021.1873988

    Article  MATH  Google Scholar 

  22. Hieu, D.V., Muu, L.D., Duong, H.N.: Iterative regularization methods for solving equilibrium problems. Inter. J. Comput. Math. (2021). https://doi.org/10.1080/00207160.2021.1906421

    Article  MATH  Google Scholar 

  23. Hieu, D.V., Quy, P.K., Duong, P.N.: Equilibrium programming and new iterative methods in Hilbert spaces. Acta Appl. Math. (2021). https://doi.org/10.1007/s10440-021-00451-0

    Article  MATH  Google Scholar 

  24. Hieu, D.V., Muu, L.D., Quy, P.K.: Regularization iterative method of bilevel form for equilibrium problems in Hilbert spaces. Math. Meth. Appl. Sci. 45, 6143–6164 (2022). https://doi.org/10.1002/mma.8162

    Article  Google Scholar 

  25. Hieu, D.V., Anh, P.K., Muu, L.D., Strodiot, J.J.: Iterative regularization methods with new stepsize rules for solving variational inclusions. J. Appl. Math. Comput. (2021). https://doi.org/10.1007/s12190-021-01534-9

    Article  MATH  Google Scholar 

  26. Hieu, D.V., Muu, L.D., Quy, P.K.: One-step optimization method for equilibrium problems. Adv. Comput. Math. 48, 29 (2022). https://doi.org/10.1007/s10444-022-09953-3

    Article  MATH  Google Scholar 

  27. Hung, P.G., Muu, L.D.: The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions. Nonlinear Analysis 74, 6121–6129 (2011)

    Article  MATH  Google Scholar 

  28. Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie Metody. 12, 747–756 (1976)

    MATH  Google Scholar 

  29. Khoroshilova, E.V.: Extragradient-type method for optimal control problem with linear constraints and convex objective function. Optim. Lett. 7, 1193–1214 (2013)

    Article  MATH  Google Scholar 

  30. Konnov, I.V.: Equilibrium Models and Variational Inequalities. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  32. Konnov, I.V., Ali, M.S.S.: Descent methods for monotone equilibrium problems in Banach spaces. J Comput. Appl. Math. 188, 165–179 (2006)

    Article  MATH  Google Scholar 

  33. Lyashko, S.I., Semenov, V.V.: Optimization and Its Applications in Control and Data Sciences. Springer, Switzerland 115, 315–325 (2016)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  35. Maingé, P.E., Moudafi, A.: Coupling viscosity methods with the extragradient algorithm for solving equilibrium problems. J. Nonlinear Convex Anal. 9, 283–294 (2008)

    MATH  Google Scholar 

  36. Martinet, B.: Régularisation d’inéquations variationelles par approximations successives. Rev. Fr. Autom. Inform. Rech. Opér., Anal. Numér. 4, 154–159 (1970)

    MATH  Google Scholar 

  37. Moudafi, A.: Proximal point algorithm extended to equilibrium problems. J. Nat. Geometry 15, 91–100 (1999)

    MATH  Google Scholar 

  38. Moudafi, A.: Proximal methods for a class of bilevel monotone equilibrium problems. J. Glob. Optim. 47, 287–292 (2010)

    Article  MATH  Google Scholar 

  39. Munkong, J., Dinh, B.V., Ungchittrakool, K.: An inertial extragradient method for solving bilevel equilibrium problems. Carpathian J. Math. 36, 91–107 (2020)

    Article  MATH  Google Scholar 

  40. 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  MATH  Google Scholar 

  41. Quoc, T.D., Muu, L.D., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  43. Sabach, S., Shtern, S.: A first-order method for solving convex bilevel optimization problems. SIAM J. Optim. 27, 640–660 (2017)

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  45. Seydenschwanz, M.: Convergence results for the discrete regularization of linear-quadratic control problems with bang-bang solutions. Comput. Optim. Appl. 629, 731–760 (2015)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  47. Xu, H.K.: Averaged mappings and the gradient-projection algorithm. J. Optim. Theory Appl. 150, 360–378 (2011)

    Article  MATH  Google Scholar 

  48. Xu, H.K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. 66, 240–256 (2002)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors sincerely thank the Editor and anonymous reviewers for their constructive comments which helped them to improve the quality and presentation of this paper. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.01-2020.06.

Author information

Authors and Affiliations

  1. TIMAS - Thang Long University, Hanoi, Vietnam

    Dang Van Hieu

  2. Department of Basic Sciences, University of Air Force, Nha Trang, Vietnam

    Pham Kim Quy

Authors
  1. Dang Van Hieu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pham Kim Quy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dang Van Hieu.

Ethics declarations

Conflict of interest

There are no conflicts of interest to this work. All authors have contributed equally.

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

Van Hieu, D., Quy, P.K. One-Step iterative method for bilevel equilibrium problem in Hilbert space. J Glob Optim 85, 487–510 (2023). https://doi.org/10.1007/s10898-022-01207-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-022-01207-2

Keywords

Mathematics Subject Classification

Search

Navigation