Optimization Letters

, Volume 12, Issue 3, pp 551–566 | Cite as

Projected subgradient algorithms on systems of equilibrium problems

  • Dang Van Hieu
Original Paper


The paper proposes two parallel and cyclic algorithms for solving systems of equilibrium problems in Hilbert spaces. The algorithms combine two methods including the diagonal subgradient method and the projection method with parallel or cyclic computations. The obtained results can be considered as improvements over several previously known methods for systems of equilibrium problems in computational steps. The algorithms have also allowed to reduce several assumptions imposed on bifunctions. The strongly convergent theorems are established under suitable conditions.


Equilibrium problem Monotone bifunction Pseudomonotone bifunction Diagonal subgradient method Projection method 



The author would like to thank the Associate Editor and the 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 also gratefully acknowledged.


  1. 1.
    Anh, P.K., Buong, N.G., Hieu, D.V.: Parallel methods for regularizing systems of equations involving accretive operators. Appl. Anal. 93, 2136–2157 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bauschke, H.H., Combettes, P.L.: A weak-to-strong convergence principle for Fejer monotone methods in Hilbert spaces. Math. Oper. Res. 26, 248–264 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–146 (1994)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90, 31–43 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Censor, Y., Gibali, A., Reich, S., Sabach, S.: Common solutions to variational inequalities. Set Valued Var. Anal. 20, 229–247 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Censor, Y., Gordon, D., Gordon, R.: Component averaging: an efficient iterative parallel algorithm for large and sparse unstructured problems. Parallel Comput. 27, 777–808 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Flam, S.D., Antipin, A.S.: Equilibrium programming and proximal-like algorithms. Math. Program. 78, 29–41 (1997)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)zbMATHGoogle Scholar
  11. 11.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hieu, D.V.: Parallel extragradient-proximal methods for split equilibrium problems. Math. Model. Anal. 21, 478–501 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hieu, D.V.: Cyclic subgradient extragradient methods for equilibrium problems. Arab. J. Math. 5, 159–175 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hieu, D.V.: Parallel hybrid methods for generalized equilibrium problems and asymptotically strictly pseudocontractive mappings. J. Appl. Math. Comput. 53, 531–554 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hieu, D.V.: An extension of hybrid method without extrapolation step to equilibrium problems. J. Ind. Manag. Optim. (2016). doi: 10.3934/jimo.2017015 Google Scholar
  17. 17.
    Hieu, D.V.: Halpern subgradient extragradient method extended to equilibrium problems. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales-Serie A: Matematicas (2016). doi: 10.1007/s13398-016-0328-9
  18. 18.
    Iiduka, H.: Convergence analysis of iterative methods for nonsmooth convex optimization over fixed point sets of quasi-nonexpansive mappings. Math. Program. 159, 509–538 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Iusem, A.N.: On some properties of paramonotone operator. Convex Anal. 5, 269–278 (1998)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Iusem, A.N., Sosa, W.: Iterative algorithms for equilibrium problems. Optimization 52(3), 301–316 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Konnov, I.V.: On systems of variational inequalities. Russ. Math. (Iz.Vuz.) 41, 79–88 (1997)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomika i Matematicheskie Metody 12, 747–756 (1976)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Maingé, P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set Valued Anal. 16, 899–912 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Martinet, B.: Régularisation d’inéquations variationelles par approximations successives. Rev. Fr. Autom. Inform. Rech. Opér., Anal. Numér. 4, 154–159 (1970)zbMATHGoogle Scholar
  25. 25.
    Moudafi, A.: Proximal point algorithm extended to equilibrum problem. J. Nat. Geom. 15, 91–100 (1999)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Mordukhovich, B.S., Panicucci, B., Pappalardo, M., Passacantando, M.: Hybrid proximal methods for equilibrium problems. Optim. Lett. 6, 1535–1550 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Muu, L.D., Oettli, W.: Convergence of an adative penalty scheme for finding constrained equilibria. Nonlinear Anal. TMA 18, 1159–1166 (1992)CrossRefzbMATHGoogle Scholar
  28. 28.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Nguyen, T.P.D., Strodiot, J.J., Nguyen, V.H., Nguyen, T.T.V.: A family of extragradient methods for solving equilibrium problems. J. Ind. Manag. Optim. 11, 619–630 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Quoc, T.D., Muu, L.D., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Santos, P., Scheimberg, S.: An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 30, 91–107 (2011)MathSciNetzbMATHGoogle Scholar
  33. 33.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Vuong, P.T., Strodiot, J.J., Nguyen, V.H.: On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space. Optimization 64, 429–451 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. Math. Anal. Appl. 298, 279–291 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Yen, L.H., Muu, L.D., Huyen, N.T.T.: An algorithm for a class of split feasibility problems: application to a model in electricity production. Math. Method Oper. Res. 84, 549–565 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsVietnam National UniversityThanh Xuan, HanoiVietnam

Personalised recommendations