A strong convergence theorem for Tseng’s extragradient method for solving variational inequality problems

  • Duong Viet Thong
  • Nguyen The Vinh
  • Yeol Je ChoEmail author
Original paper


In this paper, we introduce a new algorithm for solving variational inequality problems with monotone and Lipschitz-continuous mappings in real Hilbert spaces. Our algorithm requires only to compute one projection onto the feasible set per iteration. We prove under certain mild assumptions, a strong convergence theorem for the proposed algorithm to a solution of a variational inequality problem. Finally, we give some numerical experiments illustrating the performance of the proposed algorithm for variational inequality problems.


Tseng’s extragradient Viscosity method Variational inequality problem 



The authors would like to two anonymous reviewers for their comments on the manuscript which helped us very much in improving and presenting the original version of this paper. This paper was completed when the first two authors were visiting the Vietnam Institute for Advance Study in Mathematics (VIASM) and they thank the VIASM for financial support and hospitality and the second named author is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No.101.01-2017.08.


  1. 1.
    Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9, 3–11 (2001)MathSciNetCrossRefzbMATHGoogle 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.
    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)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, 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, 1119–1132 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dong, Q.L., Cho, Y.J., Zhong, L.L., Rassias, Th.M: Inertial projection and contraction algorithms for variational inequalities. J. Glob. Optim. 70, 687–704 (2018)Google Scholar
  10. 10.
    Dong, Q.L., Yuan, H.B., Cho, Y.J., Rassias, Th.M: Modified inertial Mann algorithm and inertial \(CQ\)-algorithm for nonexpansive mappings. Optim. Lett. 12, 87–102 (2018)Google Scholar
  11. 11.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research, vol. I. Springer, New York (2003)zbMATHGoogle Scholar
  12. 12.
    Gibali, A., Reich, S., Zalas, R.: Outer approximation methods for solving variational inequalities in Hilbert space. Optimization 66, 417–437 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)zbMATHGoogle Scholar
  14. 14.
    Harker, P.T., Pang, J.S.: A damped-Newton method for the linear complementarity problem. Lect. Appl. Math. 26, 265–284 (1990)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kimura, Y., Saejung, S.: Strong convergence for a common fixed point of two different generalizations of cutter operators. Linear Nonlinear Anal. 1, 53–65 (2015)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)zbMATHGoogle Scholar
  17. 17.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2001)CrossRefzbMATHGoogle Scholar
  19. 19.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekon. Mat. Metody 12, 747–756 (1976)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Kopecká, E., Reich, S.: Approximating fixed points in the Hilbert ball. J. Nonlinear Convex Anal. 15, 819–829 (2014)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Maingé, P.E.: Inertial iterative process for fixed points of certain quasi-nonexpansive mappings. Set-Valued Anal. 15, 67–79 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Maingé, P.E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47, 1499–1515 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Malitsky, Y.V., Semenov, V.V.: A hybrid method without extrapolation step for solving variational inequality problems. J. Glob. Optim. 61, 193–202 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Malitsky, Y.V.: Projected reflected gradient methods for monotone variational inequalities. SIAM J. Optim. 25, 502–520 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Moudafi, A.: Viscosity approximation methods for fixed points problems. J. Math. Anal. Appl. 241, 46–55 (2000)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, 191–201 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Polyak, B.T.: Some methods of speeding up the convergence of iterarive methods. Zh. Vychisl. Mat. Mat. Fiz. 4, 1–17 (1964)Google Scholar
  28. 28.
    Saejung, S., Yotkaew, P.: Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal. 75, 742–750 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Suantai, S., Pholasa, N., Cholamjiak, P.: The modified inertial relaxed CQ algorithm for solving the split feasibility problems. J. Ind. Manag. Optim. 13, 1–21 (2018). Google Scholar
  30. 30.
    Thong, D.V., Hieu, D.V.: Weak and strong convergence theorems for variational inequality problems. Numer. Algorithms 78, 1045–1060 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Thong, D.V., Hieu, D.V.: Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems. Optimization 67, 83–102 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Thong, D.V., Hieu, D.V.: Modified subgradient extragradient method for inequality variational problems. Numer. Algorithms 79, 597–610 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Thong, D.V., Hieu, D.V.: Inertial extragradient algorithms for strongly pseudomonotone variational inequalities. J. Comput. Appl. Math. 341, 80–98 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Thong, D.V., Hieu, D.V.: New extragradient methods for solving variational inequality problems and fixed point problems. J. Fixed Point Theory Appl. 20, 129 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Thong, D.V., Hieu, D.V.: Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems. Numer. Algorithms (2018). zbMATHGoogle Scholar
  36. 36.
    Tseng, P.: A modified forward–backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431–446 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wang, F.H., Xu, H.K.: Weak and strong convergence theorems for variational inequality and fixed point problems with Tseng’s extragradient method. Taiwan. J. Math. 16, 1125–1136 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Yao, Y., Marino, G., Muglia, L.: A modified Korpelevich’s method convergent to the minimum-norm solution of a variational inequality. Optimization 63, 559–569 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Wang, Y.M., Xiao, Y.B., Wang, X., Cho, Y.J.: Equivalence of well-posedness between systems of hemivariational inequalities and inclusion problems. J. Nonlinear Sci. Appl. 9, 1178–1192 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Xiao, Y.B., Huang, N.J., Cho, Y.J.: A class of generalized evolution variational inequalities in Banach space. Appl. Math. Lett. 25, 914–920 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Applied Analysis Research Group, Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam
  2. 2.Department of MathematicsUniversity of Transport and CommunicationsHanoi CityVietnam
  3. 3.Department of Mathematics Education and the RINSGyeongsang National UniversityJinjuKorea
  4. 4.School of Mathematical SciencesUniversity of Electronic Science and Technology of ChinaChengduPeople’s Republic of China

Personalised recommendations