Acta Mathematica Vietnamica

, Volume 44, Issue 2, pp 531–552 | Cite as

A Two-Step Extragradient-Viscosity Method for Variational Inequalities and Fixed Point Problems

  • Dang Van HieuEmail author
  • Dang Xuan Son
  • Pham Ky Anh
  • Le Dung Muu


This paper develops a two-step extragradient-viscosity method for finding a common solution to a variational inequality problem and a fixed point problem in a Hilbert space. Combining the two-step extragradient algorithm and the viscosity one, we propose a strongly convergent method, which contains as special cases some existing algorithms. Several numerical experiments are implemented to demonstrate the performance of the proposed method in comparison with classical hybrid extragradient-like algorithms.


Variational inequality Fixed point problem Monotone operator Demicontractive mapping Extragradient method Viscosity method Hybrid method 

Mathematics Subject Classification (2010)

65J15 47H05 47J25 47J20 91B50 



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

Funding Information

The first-named and last-named authors are supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the project: 101.01-2017.315.


  1. 1.
    Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities. Applications to Free Boundary Problems. Wiley, New York (1984)zbMATHGoogle Scholar
  2. 2.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bauschke, H.H., Chen, J., Wang, X.: A projection method for approximating fixed points of quasi nonexpansive mappings without the usual demiclosedness condition. J. Nonlinear Convex Anal. 15(1), 129–135 (2014)MathSciNetzbMATHGoogle 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(2), 318–335 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Censor, Y., Gibali, A., Reich, S.: Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim. Methods Softw. 26(4–5), 827–845 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Censor, Y., Gibali, A., Reich, S.: Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optimization 61(9), 1119–1132 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Eslamian, M., Saadati, R., Vahidi, J.: Viscosity iterative process for demicontractive mappings and multivalued mappings and equilibrium problems. Comput. Appl. Math. 36(3), 1239–1253 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Facchinei, F., Pang, J.S.: Finite-dimensional Variational Inequalities and Complementary Problems. Springer, New York (2003)zbMATHGoogle Scholar
  9. 9.
    Hartman, P., Stampacchia, G.: On some non-linear elliptic differential-functional equations. Acta Math. 115, 271–310 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hieu, D.V., Muu, L., Anh, P.K.: Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings. Numer. Algorithms 73(1), 197–217 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    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(1), 75–96 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hieu, D.V.: Parallel extragradient-proximal methods for split equilibrium problems. Math. Model. Anal. 21(4), 478–501 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hieu, D.V.: Convergence analysis of a new algorithm for strongly pseudomontone equilibrium problems. Numer. Algorithms 77(4), 983–1001 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hieu, D.V.: An extension of hybrid method without extrapolation step to equilibrium problems. J. Ind. Manag. Optim. 13(4), 1723–1741 (2017)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hieu, D.V.: An explicit parallel algorithm for variational inequalities. Bull. Malays. Math. Sci. Soc . (2017)
  16. 16.
    Hieu, D.V.: An inertial-like proximal algorithm for equilibrium problems. Math. Methods Oper Res. (2018)
  17. 17.
    Hieu, D.V., Thong, D.V.: New extragradient-like algorithms for strongly pseudomonotone variational inequalities. J. Global Optim. 70(2), 385–399 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hieu, D.V., Thong, D.V.: A new projection method for a class of variational inequalities. Appl. Anal. (2018)
  19. 19.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)zbMATHGoogle Scholar
  20. 20.
    Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2000)zbMATHGoogle Scholar
  21. 21.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie 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(2), 399–412 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Malitsky, Yu.V.: Projected reflected gradient methods for monotone variational inequalities. SIAM J. Optim. 25(1), 502–520 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Malitsky, Yu.V., Semenov, V.V.: A hybrid method without extrapolation step for solving variational inequality problems. J. Global Optim. 61(1), 193–202 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Maingé, P.E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47(3), 1499–1515 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Nadezhkina, N., Takahashi, W.: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 128(1), 191–201 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Nguyen, T.P.D., Strodiot, J.J., Nguyen, T.T.V., Nguyen, V.H.: A family of extragradient methods for solving equilibrium problems. J. Ind. Manag. Optim. 11 (2), 619–630 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118(2), 417–428 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Thong, D.V., Hieu, D.V.: Inertial extragradient algorithms for strongly pseudomonotone variational inequalities. J. Comput. Appl. Math. 341, 80–98 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Yamada, I.: The Hybrid Steepest Descent Method for the Variational Inequality Problem over the Intersection of Fixed Point Sets of Nonexpansive Mappings. In: Inherently Parallel Algorithms for Feasibility and Optimization and Their Applications, pp. 473–504, Stud. Comput. Math 8, North-Holland, Amsterdam (2001)Google Scholar
  32. 32.
    Zaporozhets, D.N., Zykina, A.V., Meleńchuk, N.V.: Comparative analysis of the extragradient methods for solution of the variational inequalities of some problems. Autom. Remote Control 73(4), 626–636 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Zeng, L.C., Yao, J.C.: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwanese J. Math. 10 (5), 1293–1303 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Zykina, A.V., Meleńchuk, N.V.: A two-step extragradient method for variational inequalities. Russian Math. (Iz. VUZ) 54(9), 71–73 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Zykina, A.V., Meleńchuk, N.V.: A doublestep extragradient method for solving a resource management problem. Model. Anal. Inform. Sys. 17(1), 65–75 (2010)Google Scholar
  36. 36.
    Zykina, A.V., Meleńchuk, N.V.: A doublestep extragradient method for solving a problem of the management of resources. Autom. Control Comput. Sci. 45(7), 452–459 (2011)CrossRefGoogle Scholar
  37. 37.
    Zeidler, E.: Nonlinear Functional Analysis and its Applications. III. Variational Methods and Optimization. Springer, New York (1985)CrossRefzbMATHGoogle Scholar
  38. 38.
    Wang, S.: Strong convergence of a regularization algorithm for common solutions with applications. Comput. Appl. Math. 35(1), 153–169 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Dang Van Hieu
    • 1
    Email author
  • Dang Xuan Son
    • 2
  • Pham Ky Anh
    • 3
  • Le Dung Muu
    • 4
  1. 1.Applied Analysis Research Group, Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam
  2. 2.VNUHanoi University of ScienceHanoiVietnam
  3. 3.Department of MathematicsVietnam National University, HanoiHanoiVietnam
  4. 4.TIMASThang Long UniversityHanoiVietnam

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