Results in Mathematics

, 74:137 | Cite as

Explicit Extragradient-Like Method with Regularization for Variational Inequalities

  • Dang Van HieuEmail author
  • Le Dung Muu
  • Pham Kim Quy
  • Le Van Vy


In this paper, we introduce and analyze the convergence of a new algorithm for solving a monotone and Lipschitz variational inequality problem in a Hilbert space. The algorithm uses variable stepsizes which are generated over each iteration, based on some previous iterates, and by some cheap computations. Contrary to many known algorithms, the resulting algorithm can be easily implemented without the prior knowledge of Lipschitz contant of operator, and also without any linesearch procedure. Besides, the regularization technique is suitably incorporated in the algorithm to get further convergence. Theorem of strong convergence is established under mild conditions imposed on control parameters. Some experiments are provided to illustrate the numerical behavior of the algorithm in comparison with others.


Variational inequality monotone operator extragradient method subgradient extragradient method projection method 

Mathematics Subject Classification

65Y05 65K15 68W10 47H05 47H10 



The authors would like to thank the Editor and the referees for their comments on the manuscript which helped in improving earlier version of this paper. The first and second authors are supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.01-2017.315.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Alber, Ya., Ryazantseva, I.: Nonlinear Ill-Posed Problems of Monotone Type. Springer, Dordrecht (2006)zbMATHGoogle Scholar
  2. 2.
    Anh, P.K., Buong, Ng, Hieu, D.V.: Parallel methods for regularizing systems of equations involving accretive operators. Appl. Anal. 93, 2136–2157 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Anh, P.K., Chung, C.V.: Parallel iterative regularization methods for solving systems of ill-posed equations. Appl. Math. Comput. 212, 542–550 (2009)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bakushinskii, A.B.: Methods for the solution of monotone variational inequalities that are based on the principle of iterative regularization. Zh. Vychisl. Mat. Mat. Fiz. 17, 1350–1362 (1977)MathSciNetGoogle Scholar
  5. 5.
    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)MathSciNetCrossRefGoogle Scholar
  6. 6.
    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)MathSciNetCrossRefGoogle Scholar
  7. 7.
    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)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dong, Q.L., Jiang, D., Gibali, A.: A modified subgradient extragradient method for solving the variational inequality problem. Numer. Algorthms 79, 927–940 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Facchinei, F., Pang, J.S.: Finite - Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2003)zbMATHGoogle Scholar
  10. 10.
    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)zbMATHGoogle Scholar
  11. 11.
    Hartman, P., Stampacchia, G.: On some non-linear elliptic diferential-functional equations. Acta Math. 115, 271–310 (1966)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hieu, D.V., Thong, D.V.: New extragradient-like algorithms for strongly pseudomonotone variational inequalities. J. Glob. Optim. 70, 385–399 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    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
  14. 14.
    Hieu, D.V., Anh, P.K., Muu, L.D.: Modified extragradient-like algorithms with new stepsizes for variational inequalities. Comput. Optim. Appl. 73, 913–932 (2019)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hieu, D.V., Cho, Y.J., Xiao, Y.-B.: Golden ratio algorithms with new stepsize rules for variational inequalities. Math. Methods Appl. Sci. (2019). CrossRefGoogle Scholar
  16. 16.
    Hieu, D.V., Quy, P.K.: An inertial modified algorithm for solving variational inequalities. RAIRO Oper. Res. (2019). CrossRefGoogle Scholar
  17. 17.
    Hieu, D.V., Thong, D.V.: A new projection method for a class of variational inequalities. Appl. Anal. (2018). CrossRefzbMATHGoogle Scholar
  18. 18.
    Hieu, D.V., Strodiot, J.J., Muu, L.D.: Strongly convergent algorithms by using new adaptive regularization parameter for equilibrium problems (2019) (submitted)Google Scholar
  19. 19.
    Khanh, P.D., Vuong, P.T.: Modified projection method for strongly pseudomonotone variational inequalities. J. Glob. Optim. 58, 341–350 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)zbMATHGoogle Scholar
  21. 21.
    Konnov, I.V.: Equilibrium Models and Variational Inequalities. Elsevier, Amsterdam (2007)zbMATHGoogle Scholar
  22. 22.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie Metody 12, 747–756 (1976)MathSciNetzbMATHGoogle Scholar
  23. 23.
    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)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Le, V.K.: Existence results for quasi-variational inequalities with multivalued perturbations of maximal monotone mappings. Results Math. 71, 423–453 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    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
  26. 26.
    Malitsky, Y.V.: Projected reflected gradient methods for monotone variational inequalities. SIAM J. Optim. 25, 502–520 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Malitsky, Y.V., Semenov, V.V.: An extragradient algorithm for monotone variational inequalities. Cybern. Syst. Anal. 50, 271–277 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Moudafi, A.: Viscosity approximations methods for fixed point problems. J. Math. Anal. Appl. 241, 46–55 (2000)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Nadezhkina, N., Takahashi, W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J. Optim. 16, 1230–1241 (2006)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Popov, L.D.: A modification of the Arrow–Hurwicz method for searching for saddle points. Mat. Zametki 28(5), 777–784 (1980)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Sun, D.: A projection and contraction method for the nonlinear complementarity problems and its extensions. Math. Numer. Sin. 16, 183–194 (1994)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Thong, D.V., Hieu, D.V.: Weak and strong convergence theorems for variational inequality problems. Numer. Algorithms 78, 1045–1060 (2018)MathSciNetCrossRefGoogle 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 80, 1283–1307 (2019)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Tinti, F.: Numerical solution for pseudomonotone variational inequality problems by extragradient methods. Var. Anal. Appl. 79, 1101–1128 (2004)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Vuong, P.T., Shehu, Y.: Convergence of an extragradient-type method for variational inequality with applications to optimal control problems. Numer. Algorithms (2018). CrossRefzbMATHGoogle Scholar
  38. 38.
    Xu, H.K.: Another control condition in an iterative method for nonexpansive mappings. Bull. Aust. Math. Soc. 65, 109–113 (2002)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Yang, J., Liu, H.: Strong convergence result for solving monotone variational inequalities in Hilbert space. Numer. Algorithms 3, 741–752 (2018). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Applied Analysis Research Group, Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam
  2. 2.TIMASThang Long UniversityHa NoiVietnam
  3. 3.Department of MathematicsCollege of Air ForceNha Trang CityVietnam

Personalised recommendations