Skip to main content
SpringerLink
Log in

Modified Extragradient Method for Variational Inequalities and Verification of Solution Existence

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

Abstract

In this paper, we propose a modified extragradient method for solving variational inequalities (VI) which has the following nice features: (i) The generated sequence possesses an expansion property with respect to the starting point; (ii) the existence of the solution to a VI problem can be verified through the behavior of the generated sequence from the fact that the iterative sequence diverges to infinity if and only if the solution set is empty. Global convergence of the method is guaranteed under mild conditions. Our preliminary computational experience is also reported.

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. Cottle, R. W., Pang, J. S., and Stone, R. E., The Linear Complementarity Problem, Academic Press, New York, NY, 1992.

    Google Scholar 

  2. Ferris, M. C., and Pang, J. S., Engineering and Economic Applications of Complementarity Problems, SIAM Review, Vol. 39, pp. 669-713, 1997.

    Google Scholar 

  3. Ferris, M. C., and Pang, J. S., Complementarity and Variational Problems: State of Art, SIAM Publications, Philadelphia, Pennsylvania, 1997.

    Google Scholar 

  4. Harker, P. T., and Pang, J. S., Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithm, and Applications, Mathematical Programming, Vol. 48, pp. 161-220, 1990.

    Google Scholar 

  5. Kinderlehrer, D., and Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, NY, 1980.

    Google Scholar 

  6. Glowinski, R., Numerical Methods for Nonlinear Variational Inequality Problems, Springer Verlag, New York, NY, 1984.

    Google Scholar 

  7. Crouzeix, J. P., Pseudomonotone Variational Inequality Problems: Existence of Solutions, Mathematical Programming, Vol. 78, pp. 305-314, 1997.

    Google Scholar 

  8. Sun, D., A Class of Iterative Methods for Solving Nonlinear Projection Equations, Journal of Optimization Theory and Applications, Vol. 91, pp. 123-140, 1996.

    Google Scholar 

  9. Korpelevich, G. M., The Extragradient Method for Finding Saddle Points and Other Problems, Matecon, Vol. 12, pp. 747-756, 1976.

    Google Scholar 

  10. Khobotov, E. N., Modification of the Extragradient Method for Solving Variational Inequalities and Certain Optimization Problem, USSR Computational Mathematics and Mathematical Physics, Vol. 27, pp. 120-127, 1987.

    Google Scholar 

  11. Sibony, M., Méthodes Itératives pour les Equations et Inéquations aux Dérivées Partielles Nonlinéares de Type Monotone, Calcolo, Vol. 7, pp. 65-183, 1970.

    Google Scholar 

  12. Bakusinskji, A. B., and Polyak, B. T., On the Solution of Variational Inequalities, Soviet Mathematics Doklady, Vol. 15, pp. 1705-1710, 1974.

    Google Scholar 

  13. He, B. S., A Class of Projection and Contraction Methods for Monotone Variational Inequalities, Applied Mathematics and Optimization, Vol. 35, pp. 69-76, 1997.

    Google Scholar 

  14. Solodov, M. V., and Tseng, P., Modified Projection-Type Methods for Monotone Variational Inequalities, SIAM Journal on Control and Optimization, Vol. 34, pp. 1814-1830, 1996.

    Google Scholar 

  15. Iusem, A. N., and Svaiter, B. F., A Variant of Korpelevich's Method for Variational Inequalities with a New Search Strategy, Optimization, Vol. 42, pp. 309-321, 1997.

    Google Scholar 

  16. Solodov, M. V., and Svaiter, B. F., A New Projection Method for Variational Inequality Problems, SIAM Journal on Control and Optimization, Vol. 37, pp. 765-776, 1999.

    Google Scholar 

  17. Konnov, I. V., A Class of Combined Iterative Methods for Solving Variational Inequalities, Vol. 94, pp. 677-693, 1997.

    Google Scholar 

  18. Xiu, N. H., Wang, C. Y., and Zhang, J. Z., Convergence Properties of Projection and Contraction Methods for Variational Inequalities Problems, Applied Mathematics and Optimization, Vol. 43, pp. 147-168, 2001.

    Google Scholar 

  19. Wang, Y. J., Xiu, N. H., and Wang, C. Y., A New Version of Extragradient Method for Variational Inequality Problems, Computers and Mathematics with Applications, Vol. 42, pp. 969-979, 2001.

    Google Scholar 

  20. Wang, Y. J., Xiu, N. H., and Wang, C. Y., Unified Framework of Extragradient-Type Methods for Pseudomonotone Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 111, pp. 641-656, 2001.

    Google Scholar 

  21. Xiu, N. H., and Zhang, J. Z., Some Recent Advances in Projection-Type Methods for Variational Inequalities, Journal of Computational and Applied Mathematics (to appear).

  22. Zarantonello, E. H., Projections on Convex Sets in Hilbert Space and Spectral Theory, Contributions to Nonlinear Functional Analysis, Edited by E. H. Zarantonello, Academic Press, New York, NY, 1971.

    Google Scholar 

  23. Solodov, M. V., and Svaiter, B. F., Forcing Strong Convergence of Proximal Point Iterations in Hilbert Space, Mathematical Programming, Vol. 87, pp. 189-202, 2000.

    Google Scholar 

  24. Pang, J. S., and Gabriel, A., NE/SQP: A Robust Algorithm for the Nonlinear Complementarity Problem, Mathematical Programming, Vol. 60, pp. 295-337, 1993.

    Google Scholar 

  25. Mangasarian, O. L., and Solodov, M. V., Nonlinear Complementarity as Unconstrained and Constrained Minimization, Mathematical Programming, Vol. 62B, pp. 277-297, 1993.

    Google Scholar 

  26. Kanzow, C., Some Equation-Based Methods for the Linear Complementarity Problem, Optimization Methods Software, Vol. 3, pp. 327-340, 1994.

    Google Scholar 

  27. Geiger, C., and Kanzow, C., On the Solution of Monotone Complementarity Problems, Computational Optimization and Applications, Vol. 5, pp. 155-172, 1996.

    Google Scholar 

  28. Mathiesen, L., An Algorithm Based on a Sequence of Linear Complementarity Problems Applied to a Walrasian Equilibrium Model: An Example, Mathematical Programming, Vol. 37, pp. 1-18, 1987.

    Google Scholar 

  29. Jiang, H. Y., and Qi, L. Q., A New Nonsmooth Equations Approach to Nonlinear Complementarity Problems, SIAM Journal on Control and Optimization, Vol. 35, pp. 178-193, 1997.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Computer Science, Nanjing Normal University, Nanjing Jiangsu, China

    Y. J. Wang (Professor)

  2. Institute of Operations Research, Qufu Normal University, Qufu, Shandong, China

    Y. J. Wang (Professor)

  3. Department of Applied Mathematics, Northern Jiaotong University, Beijing, China

    N. H. Xiu (Professor)

  4. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

    J. Z. Zhang (Professor)

Authors
  1. Y. J. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. H. Xiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. Z. Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, Y.J., Xiu, N.H. & Zhang, J.Z. Modified Extragradient Method for Variational Inequalities and Verification of Solution Existence. Journal of Optimization Theory and Applications 119, 167–183 (2003). https://doi.org/10.1023/B:JOTA.0000005047.30026.b8

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:JOTA.0000005047.30026.b8

Search

Navigation