Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

A hybrid method without extrapolation step for solving variational inequality problems

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    MATH  Google Scholar 

  2. Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)

    MATH  Google Scholar 

  3. Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities. Applications to Free Boundary Problems. Wiley, New York (1984)

    Google Scholar 

  4. Bakushinskii, A.D., Goncharskii, A.V.: Ill-Posed Problems: Theory and Applications. Kluwer Academic Publishers, Dordrecht (1994)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)

    Book  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  10. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I, II. Springer, New York (2003)

    Google Scholar 

  11. Glowinski, R., Lions, J.L., Trémolierès, R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam (1981)

    Google Scholar 

  12. Harker, P.T., Pang, J.S.: A damped-Newton method for the linear complementarity problem. Lect. Appl. Math. 26, 265–284 (1990)

    MathSciNet  Google Scholar 

  13. Iusem, A.N., Nasri, M.: Korpelevich’s method for variational inequality problems in Banach spaces. J. Glob. Optim. 50, 59–76 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  16. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)

    MATH  Google Scholar 

  17. Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2001)

    Book  MATH  Google Scholar 

  18. Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomika i Matematicheskie Metody 12(4), 747–756 (1976)

    MATH  MathSciNet  Google Scholar 

  19. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  22. Nagurney, A.: Network Economics: A Variational Inequality Approach. Kluwer, Dordrecht (1999)

    Book  Google Scholar 

  23. Nakajo, K., Takahashi, W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372–379 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Popov, L.D.: A modification of the Arrow–Hurwicz method for search of saddle points. Math. Notes 28(5), 845–848 (1980)

    Article  Google Scholar 

  25. Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  26. Solodov, M.V., Svaiter, B.F.: Forcing stong convergence of proximal point iterations in a Hilbert space. Math. Program. 87, 189–202 (2000)

    MATH  MathSciNet  Google Scholar 

  27. Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431–446 (2000)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

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

Author information

Authors and Affiliations

  1. Department of Cybernetics, Taras Shevchenko National University of Kyiv, 64/13, Volodymyrska Str., Kiev,  01601, Ukraine

    Yu. V. Malitsky & V. V. Semenov

Authors
  1. Yu. V. Malitsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. V. Semenov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Semenov.

Additional information

The research was partially supported by the Ukrainian State Fund for Fundamental Researches under Grants GP/F44/042, GP/F49/061.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Malitsky, Y.V., Semenov, V.V. A hybrid method without extrapolation step for solving variational inequality problems. J Glob Optim 61, 193–202 (2015). https://doi.org/10.1007/s10898-014-0150-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-014-0150-x

Keywords

Mathematics Subject Classification

Search

Navigation