Skip to main content

Alternating Direction Method with Self-Adaptive Penalty Parameters for Monotone Variational Inequalities

Journal of Optimization Theory and Applications volume 106, pages 337–356 (2000)Cite this article

Abstract

The alternating direction method is one of the attractive approaches for solving linearly constrained separate monotone variational inequalities. Experience on applications has shown that the number of iterations depends significantly on the penalty parameter for the system of linear constraint equations. While the penalty parameter is a constant in the original method, in this paper we present a modified alternating direction method that adjusts the penalty parameter per iteration based on the iterate message. Preliminary numerical tests show that the self-adaptive adjustment technique is effective in practice.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. Nagurney, A., and Ramanujam, P., Transportation Network Policy Modeling with Goal Targets and Generalized Penalty Functions, Transportation Science, Vol. 30, pp. 3–13, 1996.

    Google Scholar 

  2. Fortin, M., and Glowinski, R., Editors, Augmented Lagrangian Methods: Applications to the Solution of Boundary-Valued Problems, North-Holland, Amsterdam, Holland, 1983.

    Google Scholar 

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

    Google Scholar 

  4. Glowinski, R., and Le Tallec, P., Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM Studies in Applied Mathematics, Philadelphia, Pennsylvania, 1989.

  5. Gabay, D., Applications of the Method of Multipliers to Variational Inequalities, Augmented Lagrange Methods: Applications to the Solution of Boundary-Valued Problems, Edited by M. Fortin and R. Glowinski, North Holland, Amsterdam, Holland, pp. 299–331, 1983.

    Google Scholar 

  6. Gabay, D., and Mercier, B., A Dual Algorithm for the Solution of Nonlinear Variational Problems via Finite-Element Approximations, Computer and Mathematics with Applications, Vol. 2, pp. 17–40, 1976.

    Google Scholar 

  7. Lions, P. L., and Mercier, B., Splitting Algorithms for the Sum of Two Nonlinear Operators, SIAM Journal on Numerical Analysis, Vol. 16, pp. 964–979, 1979.

    Google Scholar 

  8. Tseng, P., Applications of Splitting Algorithm to Decomposition in Convex Programming and Variational Inequalities, SIAM Journal on Control and Optimization, Vol. 29, pp. 119–138, 1991.

    Google Scholar 

  9. Eckstein, J., and Fukushima, M., Some Reformulation and Applications of the Alternating Direction Method of Multipliers, Large Scale Optimization: State of the Art, Edited by W. W. Hager et al., Kluwer Academic Publishers, Amsterdam, Holland, pp. 115–134, 1994.

    Google Scholar 

  10. Fukushima, M., Application of the Alternating Direction Method of Multipliers to Separable Convex Programming Problems, Computational Optimization and Applications, Vol. 2, pp. 93–111, 1992.

    Google Scholar 

  11. Kontogiorgis, S., and Meyer, R. R., A Variable-Penalty Alternating Directions Method for Convex Optimization, Mathematical Programming, Vol. 83, pp. 29–53, 1998.

    Google Scholar 

  12. He, B. S., and Yang, H., Some Convergence Properties of a Method of Multipliers for Linearly Constrained Monotone Variational Inequalities, Operations Research Letters, Vol. 23, pp. 151–161, 1998.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Professor, State Key Lab. for Novel Software Technology, Department of Mathematics, Nanjing University, Nanjing, China

    B. S. He

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

    H. Yang

  3. PhD Student, Department of Mathematics, Nanjing University, Nanjing, China

    S. L. Wang

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. S. L. Wang
    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. & Wang, S.L. Alternating Direction Method with Self-Adaptive Penalty Parameters for Monotone Variational Inequalities. Journal of Optimization Theory and Applications 106, 337–356 (2000). https://doi.org/10.1023/A:1004603514434

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1004603514434

  • monotone variational inequalities
  • alternating direction method
  • variable penalty parameters