Journal of Global Optimization

, Volume 61, Issue 1, pp 193–202 | Cite as

A hybrid method without extrapolation step for solving variational inequality problems



In this paper, we introduce a new method for solving variational inequality problems with monotone and Lipschitz-continuous mapping in Hilbert space. The iterative process is based on well-known projection method and the hybrid (or outer approximation) method. However we do not use an extrapolation step in the projection method. The absence of one projection in our method is explained by slightly different choice of sets in the hybrid method. We prove a strong convergence of the sequences generated by our method.


Variational inequality Monotone mapping Hybrid method Projection method Strong convergence 

Mathematics Subject Classification




The authors would like to extend their gratitude towards anonymous referees whose constructive suggestions helped us to improve the presentation of this paper.


  1. 1.
    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)MATHGoogle Scholar
  2. 2.
    Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)MATHGoogle Scholar
  3. 3.
    Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities. Applications to Free Boundary Problems. Wiley, New York (1984)Google Scholar
  4. 4.
    Bakushinskii, A.D., Goncharskii, A.V.: Ill-Posed Problems: Theory and Applications. Kluwer Academic Publishers, Dordrecht (1994)Google Scholar
  5. 5.
    Bauschke, H.H., Combettes, P.L.: A weak-to-strong convergence principle for fejer-monotone methods in Hilbert spaces. Math. Oper. Res. 26(2), 248–264 (2001)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)CrossRefMATHGoogle Scholar
  7. 7.
    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)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    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)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    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)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I, II. Springer, New York (2003)Google Scholar
  11. 11.
    Glowinski, R., Lions, J.L., Trémolierès, R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam (1981)Google Scholar
  12. 12.
    Harker, P.T., Pang, J.S.: A damped-Newton method for the linear complementarity problem. Lect. Appl. Math. 26, 265–284 (1990)MathSciNetGoogle Scholar
  13. 13.
    Iusem, A.N., Nasri, M.: Korpelevich’s method for variational inequality problems in Banach spaces. J. Glob. Optim. 50, 59–76 (2011)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    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)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Khobotov, E.N.: Modification of the extragradient method for solving variational inequalities and certain optimization problems. USSR Comput. Math. Math. Phys. 27, 120–127 (1989)CrossRefGoogle Scholar
  16. 16.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)MATHGoogle Scholar
  17. 17.
    Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2001)CrossRefMATHGoogle Scholar
  18. 18.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomika i Matematicheskie Metody 12(4), 747–756 (1976)MATHMathSciNetGoogle Scholar
  19. 19.
    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)CrossRefMATHGoogle Scholar
  20. 20.
    Lyashko, S.I., Semenov, V.V., Voitova, T.A.: Low-cost modification of Korpelevich’s method for monotone equilibrium problems. Cybern. Syst. Anal. 47, 631–639 (2011)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    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)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Nagurney, A.: Network Economics: A Variational Inequality Approach. Kluwer, Dordrecht (1999)CrossRefGoogle Scholar
  23. 23.
    Nakajo, K., Takahashi, W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372–379 (2003)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Popov, L.D.: A modification of the Arrow–Hurwicz method for search of saddle points. Math. Notes 28(5), 845–848 (1980)CrossRefGoogle Scholar
  25. 25.
    Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Solodov, M.V., Svaiter, B.F.: Forcing stong convergence of proximal point iterations in a Hilbert space. Math. Program. 87, 189–202 (2000)MATHMathSciNetGoogle Scholar
  27. 27.
    Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431–446 (2000)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of CyberneticsTaras Shevchenko National University of KyivKievUkraine

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