Skip to main content

Advertisement

SpringerLink
Log in

The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We present a subgradient extragradient method for solving variational inequalities in Hilbert space. In addition, we propose a modified version of our algorithm that finds a solution of a variational inequality which is also a fixed point of a given nonexpansive mapping. We establish weak convergence theorems for both algorithms.

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

Access this article

Log in via an institution

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. Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekon. Mat. Metody 12, 747–756 (1976)

    MATH  Google Scholar 

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

    Google Scholar 

  3. 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  MathSciNet  MATH  Google Scholar 

  4. Khobotov, E.N.: Modification of the extra-gradient method for solving variational inequalities and certain optimization problems. USSR Comput. Math. Math. Phys. 27, 120–127 (1989)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Censor, Y., Gibali, A., Reich, S.: Two extensions of Korpelevich’s extragradient method for solving the variational inequality problem in Euclidean space. Technical Report (2010)

  7. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York and Basel (1984)

    MATH  Google Scholar 

  8. Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  9. Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  10. Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Browder, F.E.: Fixed point theorems for noncompact mappings in Hilbert space. Proc. Natl. Acad. Sci. USA 53, 1272–1276 (1965)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Haifa, Mt. Carmel, 31905, Haifa, Israel

    Y. Censor

  2. Department of Mathematics, The Technion—Israel Institute of Technology, 32000, Haifa, Israel

    A. Gibali & S. Reich

Authors
  1. Y. Censor
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Gibali
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Reich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Y. Censor.

Additional information

Communicated by B.T. Polyak.

This work was partially supported by Award Number R01HL070472 from the National Heart, Lung and Blood Institute. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Heart, Lung and Blood Institute or the National Institutes of Health. The third author was partially supported by the Israel Science Foundation (Grant 647/07), by the Fund for the Promotion of Research at the Technion and by the Technion President’s Research Fund.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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). https://doi.org/10.1007/s10957-010-9757-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-010-9757-3

Keywords

Search

Navigation