A self-adaptive method for pseudomonotone equilibrium problems and variational inequalities

  • Jun YangEmail author
  • Hongwei Liu


In this paper, we introduce and analyze a new algorithm for solving equilibrium problem involving pseudomonotone and Lipschitz-type bifunction in real Hilbert space. The algorithm requires only a strongly convex programming problem per iteration. A weak and a strong convergence theorem are established without the knowledge of the Lipschitz-type constants of the bifunction. As a special case of equilibrium problem, the variational inequality is also considered. Finally, numerical experiments are performed to illustrate the advantage of the proposed algorithm.


Equilibrium problem Pseudomonotone bifunction Gradient method Variational inequality 

Mathematics Subject Classification

65J15 90C33 90C25 90C52 



The authors would like to thank the Associate Editor and the anonymous referees for their valuable comments and suggestions which helped to improve the original version of this paper.


  1. 1.
    Fan, K.: A minimax Inequality and Applications, Inequalities III, pp. 103–113. Academic Press, New York (1972)Google Scholar
  2. 2.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problem. Springer, New York (2003)zbMATHGoogle Scholar
  4. 4.
    Martinet, B.: Rgularisation dinquations variationelles par approximations successives. Rev. Fr. Autom. Inform. Rech. Opr. Anal. Numr. 4, 154–159 (1970)Google Scholar
  5. 5.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2000)zbMATHGoogle Scholar
  7. 7.
    Flam, S.D., Antipin, A.S.: Equilibrium programming and proximal-like algorithms. Math. Program. 78, 29–41 (1997)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problem. Ekonomika i Matematicheskie Metody 12, 747–756 (1976)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Quoc, T.D., Muu, L.D., Hien, N.V.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dang, V.H., Duong, V.T.: New extragradient-like algorithms for strongly pseudomonotone variational inequalities. J. Glob. Optim. 70, 385–399 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dang, V.H., Yeol, J.C., Yibin, X.: Modified extragradient algorithms for solving equilibrium problems. Optimization 67, 2003–2029 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dang, V.H.: Halpern subgradient extragradient method extended to equilibrium problems. RACSAM 111, 823–840 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dang, V.H.: Convergence analysis of a new algorithm for strongly pseudomontone equilibrium problems. Numer. Algorithms 77, 983–1001 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dang, V.H.: New inertial algorithm for a class of equilibrium problems. Numer. Algorithms 80, 1413–1436 (2019)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Nguyen, T.V.: Golden ratio algorithms for solving equilibrium problems in Hilbert spaces. arXiv:1804.01829
  16. 16.
    Mastroeni, G.: On auxiliary principle for equilibrium problems. Publicatione del Dipartimento di Mathematica DellUniversita di Pisa 3, 1244–1258 (2000)Google Scholar
  17. 17.
    Daniele, P., Giannessi, F., Maugeri, A.: Equilibrium Problems and Variational Models. Kluwer, Alphen aan den Rijn (2003)CrossRefGoogle Scholar
  18. 18.
    Malitsky, Y.: Golden ratio algorithms for variational inequalities. Math. Program. (2019). CrossRefGoogle Scholar
  19. 19.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)CrossRefGoogle Scholar
  20. 20.
    Cottle, R.W., Yao, J.C.: Pseudo-monotone complementarity problems in Hilbert space. J. Optim. Theory Appl. 75, 281–295 (1992)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Phan, T.V.: On the weak convergence of the extragradient method for solving pseudo-monotone variational inequalities. J. Optim. Theory Appl. 176, 399–409 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Yang, J., Liu, H.W., Liu, Z.X.: Modified subgradient extragradient algorithms for solving monotone variational inequalities. Optimization 67, 2247–2258 (2018)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Yang, J., Liu, H.W.: A modified projected gradient method for monotone variational inequalities. J. Optim. Theory Appl. 179(1), 197–211 (2018)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Solodov, M.V., Tseng, P.: Modified projection-type methods for monotone variational inequalities. SIAM J. Control Optim. 34(5), 1814–1830 (1996)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Malitsky, YuV: Projected reflected gradient methods for variational inequalities. SIAM J. Optim. 25(1), 502–520 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Yang, J., Liu, H.W.: Strong convergence result for solving monotone variational inequalities in Hilbert space. Numer. Algorithms 80, 741–752 (2019)MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXidian UniversityXi’anChina
  2. 2.School of Mathematics and Information ScienceXianyang Normal UniversityXianyangChina

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