Numerical Algorithms

, Volume 76, Issue 1, pp 67–91 | Cite as

Splitting extragradient-like algorithms for strongly pseudomonotone equilibrium problems

Original Paper
  • 101 Downloads

Abstract

In this paper, two splitting extragradient-like algorithms for solving strongly pseudomonotone equilibrium problems given by a sum of two bifunctions are proposed. The convergence of the proposed methods is analyzed and the R-linear rate of convergence under suitable assumptions on bifunctions is established. Moreover, a noisy data case, when a part of the bifunction is contaminated by errors, is studied. Finally, some numerical experiments are given to demonstrate the efficiency of our algorithms.

Keywords

Equilibrium problem Strong pseudomonotonicity Lipschitz-type continuity Splitting-up technique Parallel computation Error estimates 

Mathematics Subject Classification

47H05 47J25 65K10 65Y05 90C25 90C33 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alizadeh, S., Moradlou, F.: A strong convergence theorem for equilibrium problems and generalized hybrid mappings. Mediterr. J. Math. 13, 379–390 (2016)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Anh, P.K., Hieu, D.V.: Parallel hybrid iterative methods for variational inequalities, equilibrium problems, and common fixed point problems. Vietnam J. Math. 44(2), 351–374 (2016)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Hieu, D.V., Muu, L.D., Anh, P.K.: Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings. Numer. Algor. 73, 197–217 (2016)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Antipin, A.S.: Gradient approach of computing fixed points of equilibrium problems. J. Glob. Optim. 24, 285–309 (2002)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Briceño-Arias, L. M.: A Douglas-Rachford splitting method for solving equilibrium problems. Nonlinear Anal. 75, 6053–6059 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63, 123–146 (1994)MathSciNetMATHGoogle Scholar
  7. 7.
    Borwein, J.M., Lewis, A.S.: Convex analysis and nonlinear optimization: theory and examples. Springer, New York (2000)CrossRefMATHGoogle Scholar
  8. 8.
    Bello Cruz, J.Y., Millán, R.D.: A direct splitting method for nonsmooth variational inequalities. J. Optim. Theory Appl. 161, 729–737 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Combettes, L.P., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)MathSciNetMATHGoogle Scholar
  10. 10.
    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
  11. 11.
    Dinh, B.V., Muu, L.D.: A projection algorithm for solving pseudomonotone equilibrium problems and it’s application to a class of bilevel equilibria. Optimization 64(3), 559–575 (2015)MathSciNetMATHGoogle Scholar
  12. 12.
    Flam, S.D., Antipin, A.S.: Equilibrium programming using proximal-like algorithms. Math. Prog. 78, 29–41 (1997)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Giannessi, F., Maugeri, A., Pardalos, P.M.: Equilibrium problems: nonsmooth optimization and variational inequality models. Kluwer, Dordrecht (2004)CrossRefMATHGoogle Scholar
  14. 14.
    Iiduka, H., Yamada, I.: An ergodic algorithm for the power-control games for CDMA data networks. J. Math. Model. Alg. 8, 1–18 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Iiduka, H.: Fixed point optimization algorithm and its application to power control in CDMA data networks. Math. Program. 133, 227–242 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Khanh, P.D., Vuong, P.T.: Modified projection method for strongly pseudomonotone variational inequalities. J. Glob. Optim. 58(2), 341–350 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Konnov, I.V.: Combined relaxation methods for variational inequalities. Springer, Berlin (2000)MATHGoogle Scholar
  18. 18.
    Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Mastroeni, G.: Gap function for equilibrium problems. J. Glob. Optim. 27, 411–426 (2004)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Moreau, J.J.: Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. France 93, 273–299 (1965)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Moudafi, A.: On the convergence of splitting proximal methods for equilibrium problems in Hilbert spaces. J. Math. Anal. Appl. 359, 508–513 (2009)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    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
  23. 23.
    Nguyen, T.T.V., Strodiot, J.J.: The interior proximal extragradient method for solving equilibrium problems. J. Glob. Optim. 44(2), 175–192 (2009)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Noor, M.A.: Some algorithms for general monotone mixed variational inequalities. Math. Comput. Model. 29(7), 1–9 (1999)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Noor, M.A.: Iterative schemes for quasimonotone mixed variational inequalities. Optimization 50, 29–44 (2001)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Plubtieng, S., Punpaeng, R.: A general iterative method for equilibrium problems and fixed points problems in Hilbert spaces. J. Math. Anal. Appl. 336, 455–469 (2007)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Quoc, T.D., Anh, P.N., Muu, L.D.: Dual extragradient algorithms extended to equilibrium problems. J. Glob. Optim. 52(1), 139–159 (2012)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Quoc, T.D., Muu, L.D., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57(6), 749–776 (2008)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Rockafellar, R.T.: Convex analysis. Princeton University Press, Princeton, NJ (1970)CrossRefMATHGoogle Scholar
  30. 30.
    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
  31. 31.
    Svaiter, B.F.: On weak convergence of the Douglas-Rachford method. SIAM J. Control Optim. 49, 280–287 (2011)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Takahashi, S., Takahashi, W.: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 69, 1025–1033 (2008)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    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(2), 605–627 (2012)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Xu, H.K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. 66(2), 240–256 (2002)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsVietnam National UniversityHanoiVietnam
  2. 2.School of Applied Mathematics and InformaticsHanoi University of Science and TechnologyHanoiVietnam

Personalised recommendations