Numerical Algorithms

, Volume 77, Issue 4, pp 983–1001 | Cite as

Convergence analysis of a new algorithm for strongly pseudomontone equilibrium problems

Original Paper

Abstract

The paper introduces and analyzes the convergence of a new iterative algorithm for approximating solutions of equilibrium problems involving strongly pseudomonotone and Lipschitz-type bifunctions in Hilbert spaces. The algorithm uses a stepsize sequence which is non-increasing, diminishing, and non-summable. This leads to the main advantage of the algorithm, namely that the construction of solution approximations and the proof of its convergence are done without the prior knowledge of the modulus of strong pseudomonotonicity and Lipschitz-type constants of bifunctions. The strongly convergent theorem is established under suitable assumptions. The paper also discusses the assumptions used in the formulation of the convergent theorem. Several numerical results are reported to illustrate the behavior of the algorithm with different sequences of stepsizes and also to compare it with others.

Keywords

Proximal-like method Extragradient method Equilibrium problem Strongly pseudomonotone bifunction Lipschitz-type bifunction 

Mathematics Subject Classification (2010)

65J15 47H05 47J25 47J20 91B50 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The author would like to thank the Associate Editor and two anonymous referees for their valuable comments and suggestions which helped us very much in improving the original version of this paper. The guidance of Profs. P. K. Anh and L. D. Muu is gratefully acknowledged.

References

  1. 1.
    Antipin, A.S.: On convergence of proximal methods to fixed points of extremal mappings and estimates of their rate of convergence. Comp. Maths. Math. Phys. 35, 539–551 (1995)MATHGoogle Scholar
  2. 2.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)CrossRefMATHGoogle Scholar
  3. 3.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Program. 63, 123–145 (1994)MathSciNetMATHGoogle Scholar
  4. 4.
    Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6(1), 117–136 (2005)MathSciNetMATHGoogle Scholar
  5. 5.
    Contreras, J., Klusch, M., Krawczyk, J.B.: Numerical solutions to Nash-Cournot equilibria in coupled constraint electricity markets. IEEE Trans. Power Syst. 19(1), 195–206 (2004)CrossRefGoogle Scholar
  6. 6.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2002)MATHGoogle Scholar
  7. 7.
    Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequality, III, pp 103–113. Academic Press, New York (1972)Google Scholar
  8. 8.
    Flam, S.D., Antipin, A.S.: Equilibrium programming and proximal-like algorithms. Math. Program. 78, 29–41 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hieu, D.V., Muu, L.D, Anh, P.K.: Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings. Numer. Algorithms 73, 197–217 (2016)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hieu, D.V., Anh, P.K., Muu, L.D.: Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput. Optim. Appl. 66, 75–96 (2017)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hieu, D.V.: An extension of hybrid method without extrapolation step to equilibrium problems. J. Ind. Manag. Optim. (2016). doi: 10.3934/jimo.2017015
  12. 12.
    Hieu, D.V.: Halpern subgradient extragradient method extended to equilibrium problems. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM. (2016). doi: 10.1007/s13398-016-0328-9
  13. 13.
    Hieu, D.V.: Hybrid projection methods for equilibrium problems with non-Lipschitz type bifunctions. Math. Meth. Appl. Sci. (2017). doi: 10.1002/mma.4286
  14. 14.
    Hieu, D.V.: Parallel extragradient-proximal methods for split equilibrium problems. Math. Model. Anal. 21, 478–501 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Konnov, I.V.: Application of the proximal point method to nonmonotone equilibrium problems. J. Optim. Theory Appl. 119, 317–333 (2003)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Konnov, I.V.: Equilibrium Models and Variational Inequalities. Elsevier, Amsterdam (2007)MATHGoogle Scholar
  17. 17.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie Metody 12, 747–756 (1976)MathSciNetMATHGoogle Scholar
  18. 18.
    Lyashko, S.I., Semenov, V.V.: Optimization and Its Applications in Control and Data Sciences, vol. 115, pp 315–325. Springer, Switzerland (2016)CrossRefGoogle Scholar
  19. 19.
    Mastroeni, G.: On auxiliary principle for equilibrium problems, vol. 3, pp. 1244–1258. Publicatione del Dipartimento di Mathematica dell, Universita di Pisa (2000)Google Scholar
  20. 20.
    Mastroeni, G.: Gap function for equilibrium problems. J. Global. Optim. 27, 411–426 (2003)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Martinet, B.: Régularisation d’inéquations variationelles par approximations successives. Rev. Fr. Autom. Inform. Rech. Opér., Anal. Numér. 4, 154–159 (1970)MATHGoogle Scholar
  22. 22.
    Moudafi, A.: Proximal point algorithm extended to equilibrium problem. J. Nat. Geometry 15, 91–100 (1999)MathSciNetMATHGoogle Scholar
  23. 23.
    Muu, L.D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. TMA 18(12), 1159–1166 (1992)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    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)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Muu, L.D., Quy, N.V.: On existence and solution methods for strongly pseudomonotone equilibrium problems. Vietnam J. Math. 43, 229–238 (2015)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Popov, L.D.: A modification of the Arrow-Hurwicz method for search of saddle points. Math. Notes Acad. Sci. USSR 28(5), 845–848 (1980)MATHGoogle Scholar
  27. 27.
    Quoc, T.D., Muu, L.D., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)Google Scholar
  30. 30.
    Santos, P., Scheimberg, S.: An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 30, 91–107 (2011)MathSciNetMATHGoogle Scholar
  31. 31.
    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)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    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)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsVietnam National UniversityHanoiVietnam
  2. 2.Department of MathematicsCollege of Air ForceNhatrangVietnam

Personalised recommendations