Strong convergence result for solving monotone variational inequalities in Hilbert space

Original Paper


In this paper, we study strong convergence of the algorithm for solving classical variational inequalities problem with Lipschitz-continuous and monotone mapping in real Hilbert space. The algorithm is inspired by Tseng’s extragradient method and the viscosity method with a simple step size. A strong convergence theorem for our algorithm is proved without any requirement of additional projections and the knowledge of the Lipschitz constant of the mapping. Finally, we provide some numerical experiments to show the efficiency and advantage of the proposed algorithm.


Variational inequalities Projection Extragradient method Monotone mapping Convex set 

Mathematics Subject Classification (2010)

47J20 90C25 90C30 90C52 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



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.
    Hartman, P., Stampacchia, G.: On some linear elliptic differential equations. Acta Math. 115, 271–310 (1966)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problem. Ekonomika i Matematicheskie Metody 12, 747–756 (1976)MathSciNetMATHGoogle Scholar
  3. 3.
    Noor, M.A.: Some developments in general variational inequalities. Appl. Math. Comput. 152, 199–277 (2004)MathSciNetMATHGoogle Scholar
  4. 4.
    Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148, 318–335 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Malitsky, Yu.V., Semenov, V.V.: An extragradient algorithm for monotone variational inequalities. Cybern. Syst. Anal. 50, 271–277 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Tseng, P.: A modified forward-backward splitting method for maximal monotone mapping. SIAM J. Control. Optim. 38, 431–446 (2000)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Solodov, M.V., Svaiter, B.F.: A new projection method for monotone variational inequalities. SIAM. J. Control. Optim. 37, 765–776 (1999)CrossRefMATHGoogle Scholar
  8. 8.
    Malitsky, Yu.V.: Projected reflected gradient methods for variational inequalities. SIAM J. Optim. 25(1), 502–520 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Mainge, P.E., Gobinddass, M.L.: Convergence of one-step projected gradient methods for variational inequalities. J. Optim. Theory Appl. 171, 146–168 (2016)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Iusem, A.N., Svaiter, B.F.: A variant of Korpelevich’s method for variational inequalities with a new search strategy. Optimization 42, 309–321 (1997)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Duong, V.T., Dang V.H.: Weak and strong convergence theorems for variational inequality problems. Numer Algor,
  12. 12.
    Mainge, P.E.: The viscosity approximation process for quasi-nonexpansive mapping in Hilbert space. Comput. Math. Appl. 59, 74–79 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Rapeepan, K., Satit, S.: Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 163, 399–412 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Yekini, S., Olaniyi, S.I.: Strong convergence result for monotone variational inequalities. Numer Algor 76, 259–282 (2017)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gibali, A.: A new non-Lipschitzian method for solving variational inequalities in Euclidean spaces. J. Nonlinear Anal. Optim. 6, 41–51 (2015)MathSciNetGoogle Scholar
  16. 16.
    Moudafi, A.: Viscosity methods for fixed points problems. Journal of Mathematical Analysis and Applications. 241, 46–55 (2000)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Antipin, A.S.: On a method for convex programs using a symmetrical modification of the Lagrange function. Ekonomika i Matematicheskie Metody. 12(6), 1164–1173 (1976)Google Scholar
  18. 18.
    Xu, H.K.: Iterative algorithm for nonlinear operators. J. Lond. Math. Soc. 66 (2), 240–256 (2002)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

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

Personalised recommendations