Analysis and Mathematical Physics

, Volume 9, Issue 4, pp 2203–2225 | Cite as

Two simple projection-type methods for solving variational inequalities

  • Aviv Gibali
  • Duong Viet ThongEmail author
  • Pham Anh Tuan
Part of the following topical collections:
  1. Perspectives in Modern Analysis


In this paper we study a classical monotone and Lipschitz continuous variational inequality in real Hilbert spaces. Two projection type methods, Mann and its viscosity generalization are introduced with their strong convergence theorems. Our methods generalize and extend some related results in the literature and their main advantages are: the strong convergence and the adaptive step-size usage which avoids the need to know apriori the Lipschitz constant of variational inequality associated operator. Primary numerical experiments in finite and infinite dimensional spaces compare and illustrate the behaviors of the proposed schemes.


Projection-type method Variational inequality Mann-type method Viscosity method Projection and contraction method 

Mathematics Subject Classification

47H09 47J20 65K15 90C25 


Compliance with ethical standards

Conflict of interest

The authors declare no conflict of interest.


  1. 1.
    Antipin, A.S.: On a method for convex programs using a symmetrical modification of the Lagrange function. Ekonomika i Mat. Metody. 12, 1164–1173 (1976)Google Scholar
  2. 2.
    Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)zbMATHGoogle Scholar
  3. 3.
    Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities, Applications to Free Boundary Problems. Wiley, New York (1984)zbMATHGoogle Scholar
  4. 4.
    Cai, X., Gu, G., He, B.: On the \(O(1/t)\) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators. Comput. Optim. Appl. 57, 339–363 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces. Lecture Notes in Mathematics, vol. 2057. Springer, Berlin (2012)zbMATHGoogle Scholar
  6. 6.
    Ceng, L.C., Hadjisavvas, N., Wong, N.C.: Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J. Glob. Optim. 46, 635–646 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Censor, Y., Gibali, A., Reich, S.: The subgradient extragradientmethod for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148, 318–335 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Censor, Y., Gibali, A., Reich, S.: Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim. Meth. Softw. 26, 827–845 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Censor, Y., Gibali, A., Reich, S.: Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optimization 61, 1119–1132 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 56, 301–323 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dong, Q.L., Gibali, A., Jiang, D., Ke, S.H.: Convergence of projection and contraction algorithms with outer perturbations and their applications to sparse signals recovery. J. Fixed Point Theory Appl. 20, 16 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dong, L.Q., Cho, J.Y., Zhong, L.L., Rassias, MTh: Inertial projection and contraction algorithms for variational inequalities. J. Glob. Optim. 70, 687–704 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research, vols. I and II. Springer, New York (2003)Google Scholar
  14. 14.
    Fichera, G.: Sul problema elastostatico di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei, VIII Ser. Rend. Cl. Sci. Fis. Mat. Nat. 34, 138–142 (1963)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Fichera, G.: Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei, Mem. Cl. Sci. Fis. Mat. Nat. Sez. I, VIII. Ser. 7, 91–140 (1964)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)zbMATHGoogle Scholar
  17. 17.
    He, B.S.: A class of projection and contraction methods for monotone variational inequalities. Appl. Math. Optim. 35, 69–76 (1997)MathSciNetCrossRefGoogle Scholar
  18. 18.
    He, B.S., Liao, L.Z.: Improvements of some projection methods for monotone nonlinear variational inequalities. J. Optim. Theory Appl. 112, 111–128 (2002)MathSciNetCrossRefGoogle Scholar
  19. 19.
    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)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2001)CrossRefGoogle Scholar
  21. 21.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomika i Mat. Metody. 12, 747–756 (1976)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kraikaew, R., Saejung, S.: Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 163, 399–412 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Liu, L.S.: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach space. J. Math. Anal. Appl. 194, 114–125 (1995)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Maingé, P.E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47, 1499–1515 (2008)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Malitsky, Y.V.: Projected reflected gradient methods for monotone variational inequalities. SIAM J. Optim. 25, 502–520 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Malitsky, Y.V., Semenov, V.V.: A hybrid method without extrapolation step for solving variational inequality problems. J. Glob. Optim. 61, 193–202 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Moudafi, A.: Viscosity approximating methods for fixed point problems. J. Math. Anal. Appl. 241, 46–55 (2000)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Reich, S.: Constructive Techniques for Accretive and Monotone Operators. Applied Nonlinear Analysis, pp. 335–345. Academic Press, New York (1979)zbMATHGoogle Scholar
  30. 30.
    Shehu, Y., Iyiola, O.S.: Strong convergence result for monotone variational inequalities. Numer. Algorithms 76, 259–282 (2017)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Sun, D.F.: A class of iterative methods for solving nonlinear projection equations. J. Optim. Theory Appl. 91, 123–140 (1996)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Thong, D.V., Hieu, D.V.: Weak and strong convergence theorems for variational inequality problems. Numer. Algorithms 78, 1045–1060 (2018)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Thong, D.V., Hieu, D.V.: Modified subgradient extragradient method for variational inequality problems. Numer. Algorithms 79, 597–610 (2018)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Thong, D.V., Hieu, D.V.: Inertial extragradient algorithms for strongly pseudomonotone variational inequalities. J. Comput. Appl. Math. 341, 80–98 (2018)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Aviv Gibali
    • 1
    • 2
  • Duong Viet Thong
    • 3
    Email author
  • Pham Anh Tuan
    • 4
  1. 1.Department of MathematicsORT Braude CollegeKarmielIsrael
  2. 2.The Center for Mathematics and Scientific ComputationUniversity of HaifaMt. Carmel, HaifaIsrael
  3. 3.Applied Analysis Research Group, Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam
  4. 4.Faculty of Economics MathematicsNational Economics UniversityHanoi CityVietnam

Personalised recommendations