Skip to main content

Advertisement

SpringerLink
Log in

A Modified Extragradient Method for Infinite-Dimensional Variational Inequalities

  • Published:
Acta Mathematica Vietnamica Aims and scope Submit manuscript

Abstract

A modified form of the extragradient method for solving infinite-dimensional variational inequalities is considered. The weak convergence and the strong convergence for the iterative sequence generated by this method are studied. We also propose several examples to analyze the obtained results.

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. Ceng, L.C., Huang, S., Petrusel, A.: Weak convergence theorem by a modified extragradient method for nonexpansive mappings and monotone mappings. Taiwanese J. Math. 13, 225–238 (2009)

    MathSciNet  MATH  Google Scholar 

  2. 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  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  4. Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland, Amsterdam (1979)

    MATH  Google Scholar 

  5. 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 

  6. Khobotov, E.N.: Modification of the extra-gradient method for solving variational inequalities and certain optimization problems. (Russian) Zh. Vychisl. Mat. i Mat. Fiz. 27, 1462–1473 (1987)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  8. Korpelevich, G.M.: An extragradient method for finding saddle points and for other problems. (Russian) Èkonom. i Mat. Metody. 12, 747–756 (1976)

    MathSciNet  MATH  Google Scholar 

  9. Marcotte, P.: Application of Khobotov’s algorithm to variational inequalities and network equilibrium problems. Inform. Systems Oper. Res. 29, 258–270 (1991)

    Article  MATH  Google Scholar 

  10. Martinet, B.: Régularisation d’inéquations variationnelles par approximations successives. Revue Française Automatique Informatique et Recherche Opérationelle 4, 154–158 (1970)

    MathSciNet  MATH  Google Scholar 

  11. Noor, M.A.: Some algorithms for general monotone mixed variational inequalities. Math. Comput. Modelling 29, 1–9 (1999)

    MathSciNet  MATH  Google Scholar 

  12. Noor, M.A.: Some developments in general variational inequalities. Appl. Math. Comput. 152, 199–277 (2004)

    MathSciNet  MATH  Google Scholar 

  13. 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 

  14. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM. J. Control Optimization 14, 877–898 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  15. Solodov, M.V., Tseng, P.: Modified projection-type methods for monotone variational inequalities. SIAM J. Control Optimization 34, 1814–1830 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Tam, N.N., Yao, J.-C., Yen, N.D.: Solution methods for pseudomonotone variational inequalities. J. Optim. Theory Appl. 138, 253–273 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Thanh Hao, N.: Tikhonov regularization algorithm for pseudomonotone variational inequalities. Acta Math. Vietnam. 31, 283–289 (2006)

    MathSciNet  MATH  Google Scholar 

  19. Tinti, F.: Numerical solution for pseudomonotone variational inequality problems by extragradient methods. Variational analysis and applications. Nonconvex Optim. Appl. 79, 1101–1128 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Yao, J.-C.: Multi-valued variational inequalities with K-pseudomonotone operators. J. Optim. Theory Appl. 83, 391–403 (1994)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The guidance of Prof. N. D. Yen and Dr. T. C. Dieu is gratefully acknowledged. The author would like to thank the two anonymous referees for valuable remarks and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Ho Chi Minh City University of Pedagogy, 280 An Duong Vuong, Ho Chi Minh, Vietnam

    Pham Duy Khanh

Authors
  1. Pham Duy Khanh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pham Duy Khanh.

Additional information

The author is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2014.56.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khanh, P.D. A Modified Extragradient Method for Infinite-Dimensional Variational Inequalities. Acta Math Vietnam 41, 251–263 (2016). https://doi.org/10.1007/s40306-015-0150-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40306-015-0150-z

Keywords

Mathematics Subject Classification (2010)

Search

Navigation