Skip to main content
SpringerLink
Log in

Modified Goldstein–Levitin–Polyak Projection Method for Asymmetric Strongly Monotone Variational Inequalities

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

Abstract

In this paper, we present a modified Goldstein–Levitin–Polyak projection method for asymmetric strongly monotone variational inequality problems. A practical and robust stepsize choice strategy, termed self-adaptive procedure, is developed. The global convergence of the resulting algorithm is established under the same conditions used in the original projection method. Numerical results and comparison with some existing projection-type methods are given to illustrate the efficiency of the proposed method.

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. NAGURNEY, A., Network Economics: A Variational Inequality Approach, Kluwer Academic Publishers, Boston, Massachusetts, 1993.

    Google Scholar 

  2. DAFERMOS, D., An Iterative Scheme for Variational Inequalities, Mathematical Programming, Vol. 26, pp. 40-47, 1983.

    Google Scholar 

  3. FUKUSHIMA, M., Equivlent Differentiable Optimization Problems and Descent Methods for Asymmetric Variational Inequality Problems, Mathematical Programming, Vol. 53, pp. 99-110, 1992.

    Google Scholar 

  4. KANZOW, C., Nonlinear Complementarity as Unconstrained Optimization, Journal of Optimization Theory and Applications, Vol. 88, pp. 139-155, 1996.

    Google Scholar 

  5. MARCOTTE, P., A New Algorithm for Solving Variational Inequalities with Application to the Traffic Assignment Problem, Mathematical Programming, Vol. 33, pp. 339-351, 1985.

    Google Scholar 

  6. HARKER, P. T., and PANG, P. S., Finite-Dimensional Variational Inequalities and Complementarity Problems: A Survey of Theory, Algorithms, and Applications, Mathematical Programming, Vol. 48, pp. 161-220, 1990.

    Google Scholar 

  7. NOOR, M. A., Some Recent Advances in Variational Inequalities, Part 1: Basic Concepts, New Zealand Journal of Mathematics, Vol. 26, pp. 53-80, 1997.

    Google Scholar 

  8. LUO, Z. Q., Error Bounds and Convergence Analysis of Feasible Descent Methods: A General Approach, Annals of Operations Research, Vol. 46, pp. 157-178, 1993.

    Google Scholar 

  9. GOLDSTEIN, A. A., Convex Programming in Hilbert Space, Bulletin of the American Mathematical Society, Vol. 70, pp. 709-710, 1964.

    Google Scholar 

  10. LEVITIN, E. S., and POLYAK, B. T., Constrained Minimization Problems, USSR Computational Mathematics and Mathematical Physics, Vol. 6, pp. 1-50, 1966.

    Google Scholar 

  11. DUPUIS, P., and NAGURNEY, A., Dynamical Systems and Variational Inequalities, Annals of Operations Research, Vol. 44, pp. 9-42, 1993.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  14. BERTSEKAS, D. P., On the Goldstein-Levitin-Polyak Gradient Projection Method, IEEE Transactions on Automatic Control, Vol. 21, pp. 174-184, 1976.

    Google Scholar 

  15. ARMIJO, L., Minimization of Functions Having Continuous Partial Derivatives, Pacific Journal of Mathematics, Vol. 16, pp. 1-3, 1966.

    Google Scholar 

  16. BERTSEKAS, D. P., and TSITSIKLIS, J. N., Parallel Distributed Computation: Numerical Methods, Prentice-Hall, Englewood Cliffs, New Jersey, 1989.

    Google Scholar 

  17. GAFNI, E. M., and BERTSEKAS, D. P., Two-Metric Projection Methods for Constrained Optimization, SIAM Journal of Control and Optimization, Vol. 22, pp. 936-964, 1984.

    Google Scholar 

  18. HE, B. S., Some Predictor-Corrector Methods for Monotone Variational Inequalities, Technical Report 95-65, Faculty of Technical Mathematics and Informatics, TU Delft, Delft, Netherlands, 1995.

    Google Scholar 

  19. PENG, J. M., Equivalence of Variational Inequality Problems to Unconstrained Minimization, Mathematical Programming, Vol. 78, pp. 347-355, 1997.

    Google Scholar 

  20. NAGURNEY, A., and ZHANG, D., Projected Dynamical Systems and Variational Inequalities with Applications, Kluwer Academic Publishers, Boston, Massachusetts, 1996.

    Google Scholar 

  21. ZHANG, D., and NAGURNEY, A., Formulation, Stability, and Computation of Traffic Network Equilibria as Projected Dynamical Systems, Journal of Optimization Theory and Applications, Vol. 93, pp. 417-444, 1997.

    Google Scholar 

  22. MARCOTTE, P., Application of Khoboto'vs Algorithm to Variational Inequalities and Network Equilibrium Problems, Information Systems and Operations Research, Vol. 29, pp. 258-270, 1991.

    Google Scholar 

  23. HARKER, P. T., and PANG, J. S., A Damped Newton Method for the Linear Complementarity Problem, Lectures in Applied Mathematics, Vol. 26, pp. 265-284, 1990.

    Google Scholar 

  24. MARCOTE, P., and DUSSAULT, J. P., A Note on a Globally Convergent Newton Method for Solving Variational Inequalities, Operation Research Letters, Vol. 6, pp. 35-42, 1987.

    Google Scholar 

  25. TAJI, K., FUKUSHIMA, M., and IBARAKI, T., A Globally Convergent Newton Method for Solving Strongly Monotone Variational Inequalities, Mathematical Programming. Vol. 58, pp. 369-383, 1993.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, PRC

    B. S. He (Professor)

  2. Department of Civil Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, PRC

    H. Yang (Associate Professor)

  3. Department of Civil Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, PRC

    Q. Meng (PhD Student)

  4. Department of Mathematics, Nanjing University, Nanjing, PRC

    D. R. Han (PhD Student)

Authors
  1. B. S. He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Q. Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. D. R. Han
    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

He, B.S., Yang, H., Meng, Q. et al. Modified Goldstein–Levitin–Polyak Projection Method for Asymmetric Strongly Monotone Variational Inequalities. Journal of Optimization Theory and Applications 112, 129–143 (2002). https://doi.org/10.1023/A:1013048729944

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1013048729944

Search

Navigation