Journal of Global Optimization

, Volume 64, Issue 1, pp 179–195 | Cite as

On ergodic algorithms for equilibrium problems

  • P. N. Anh
  • T. N. Hai
  • P. M. Tuan


In this paper, we present a new iteration method for solving monotone equilibrium problems. This new method is based on the ergodic iteration method Ronald and Bruck in (J Math Anal Appl 61:159–164, 1977) and the auxiliary problem principle Noor in (J Optim Theory Appl 122:371–386, 2004), but it includes the usage of symmetric and positive definite matrices. The proposed algorithm is very simple. Moreover, it simplifies the assumptions necessary in order to converge to the solution. Specifically, whereas previous methods require strong monotonicity and Lipschitz-type continuous conditions, our proposed method only requires weak monotonicity conditions. Applications to the generalized variational inequality problem and some numerical results are reported.


Equilibrium problem Monotone Ergodic algorithm Auxiliary problem principle 

Mathematics Subject Classification

65 K10 90 C25 



We are grateful to the anonymous referees for their really helpful and constructive comments that helped us very much to improve the original version of the paper substantially.


  1. 1.
    Anh, P.N.: A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization 62, 271–283 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Anh, P.N.: Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities. J. Optim. Theory Appl. 154, 303–320 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Anh, P.N.: A logarithmic quadratic regularization method for solving pseudomonotone equilibrium problems. Acta Math. Vietnam. 34, 183–200 (2009)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Anh, P.N.: An LQP regularization method for equilibrium problems on polyhedral. Vietnam. J. Math. 36, 209–228 (2008)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Anh, P.N., Hien, N.D.: Fixed point solution methods for solving equilibrium problems. B. Korean Math. Soc. 51, 479–499 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Anh, P.N., Kim, J.K.: Outer approximation algorithms for pseudomonotone equilibrium problems. Comp. Math. Appl. 61, 2588–2595 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Anh, P.N., Kim, J.K., Muu, L.D.: An extragradient method for solving bilevel variational inequalities. J. Glob. Optim. 52, 627–639 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Anh, P.N., Le Thi, H.A.: An Armijo-type method for pseudomonotone equilibrium problems and Its applications. J. Glob. Optim. 57, 803–820 (2013)zbMATHCrossRefGoogle Scholar
  9. 9.
    Bigi, G., Castellani, M., Pappalardo, M.: A new solution method for equilibrium problems. Optim. Method Softw. 24, 895–911 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Blum, E., Oettli, W.: From optimization and variational inequality to equilibrium problems. Math. Stud. 63, 127–149 (1994)MathSciNetGoogle Scholar
  11. 11.
    Cho, Y.J., Qin, X., Kang, J.I.: Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems. Nonlinear Anal. 71, 4203–4214 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Cho, Y.J., Qin, X.: Systems of generalized nonlinear variational inequalities and its projection methods. Nonlinear Anal. 69, 4443–4451 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Cohen, G.: Auxiliary problem principle and decomposition of optimization problems. J. Optim. Theory Appl. 32, 277–305 (1980)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Cohen, G.: Auxiliary principle extended to variational inequalities. J. Optim. Theory Appl. 59, 325–333 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Iiduka, H., Yamada, I.: An ergodic algorithm for the power-control games for CDMA data networks. J. Math. Model. Algorithms 8, 1–18 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Iusem, A.N., Sosa, W.: On the proximal point method for equilibrium problems in Hilbert spaces. Optimization 59, 1259–1274 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Iusem, A.N., Sosa, W.: Iterative algorithms for equilibrium problems. Optimization 52, 301–316 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Kim, J.K., Anh, P.N., Hyun, H.G.: A proximal point-type algorithm for pseudomonotone equilibrium problems. Bull. Korean Math. Society 49, 749–759 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Konnov, I.V.: Application of the proximal point method to nonmonotone equilibrium problems. J. Optim. Theory Appl. 119, 317–333 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2000)Google Scholar
  21. 21.
    Mastroeni, G.: On auxiliary principle for equilibrium problems. In: Daniele, P., Giannessi, F., Maugeri, A. (eds.) Nonconvex Optimization and its Applications. Kluwer Academic Publishers, Dordrecht (2003)Google Scholar
  22. 22.
    Mastroeni, G.: Gap function for equilibrium problems. J. Global. Optim. 27, 411–426 (2004)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Moudafi, A.: Proximal point algorithm extended to equilibrium problem. J. Nat. Geom. 15, 91–100 (1999)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Noor, M.A.: Auxiliary principle technique for equilibrium problems. J. Optim. Theory Appl. 122, 371–386 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Quoc, T.D., Anh, P.N., Muu, L.D.: Dual extragradient algorithms to equilibrium Problems. J. Glob. Optim. 52, 139–159 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Ronald, E., Bruck, R.E.: On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space. J. Math. Anal. Appl. 61, 159–164 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Ronald, E., Bruck, R.E.: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Isr. J. Math. 32, 107–116 (1979)zbMATHCrossRefGoogle Scholar
  28. 28.
    Tran, Q.D., Muu, L.D., Hien, N.V.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Scientific FundamentalsPosts and Telecommunications Institute of TechnologyHanoiVietnam
  2. 2.School of Applied Mathematics and InformaticsHa Noi University of Science and TechnologyHanoiVietnam
  3. 3.Academy of Military Science and TechnologyHanoiVietnam

Personalised recommendations