Skip to main content
SpringerLink
Log in

New NCP-Functions and Their Properties

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

Abstract

Recently, Luo and Tseng proposed a class of merit functions for the nonlinear complementarity problem (NCP) and showed that it enjoys several interesting properties under some assumptions. In this paper, adopting a similar idea to that of Luo and Tseng, we present new merit functions for the NCP, which can be decomposed into component functions. We show that these merit functions not only share many properties with the one proposed by Luo and Tseng but also enjoy additional favorable properties owing to their decomposable structure. In particular, we present fairly mild conditions under which these merit functions have bounded level sets.

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. 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 

  2. PANG, J. S., Complementarity Problems, Handbook of Global Optimization, Edited by R. Horst and P. Pardalos, Kluwer Academic Publishers, Boston, Massachusetts, pp. 271–338. 1994.

    Google Scholar 

  3. FUKUSHIMA, M., Merit Functions for Variational Inequality and Complementarity Problems, Nonlinear Optimization and Applications, Edited by G. Di Pillo and F. Giannessi, Plenum Publishing Corporation, New York, New York, pp. 155–170, 1996.

    Google Scholar 

  4. FISCHER, A., An NCP-Function and Its Use for the Solution of Complementarity Problems, Recent Advances in Nonsmooth Optimization, Edited by D. Z. Du, L. Qi, and R. S. Womersley, World Scientific Publishers, Singapore, pp. 88–105, 1995.

    Google Scholar 

  5. GEIGER, C., and KANZOW, C., On the Resolution of Monotone Complementarity Problems, Computational Optimization and Applications, Vol. 5, pp. 155–173, 1996.

    Google Scholar 

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

    Google Scholar 

  7. KANZOW, C., and KLEINMICHEL, H., A Class of Newton-Type Methods for Equality and Inequality Constrained Optimization, Optimization Methods and Software, Vol. 5, pp. 173–198, 1995.

    Google Scholar 

  8. MANGASARIAN, O. L., and SOLODOV, M. V., Nonlinear Complementarity as Unconstrained and Constrained Minimization, Mathematical Programming, Vol. 62, pp. 277–297, 1993.

    Google Scholar 

  9. FISCHER, A., A Special Newton-Type Optimization Method, Optimization, Vol. 24, pp. 269–284, 1992.

    Google Scholar 

  10. LUO, Z. Q., MANGASARIAN, O. L., REN, J., and SOLODOV, M. V., New Error Bounds for the Linear Complementarity Problem, Mathematics of Operations Research, Vol. 19, pp. 880–892, 1994.

    Google Scholar 

  11. TSENG, P., Growth Behavior of a Class of Merit Functions for the Nonlinear Complementarity Problem, Journal of Optimization Theory and Applications, Vol. 89, pp. 17–37, 1996.

    Google Scholar 

  12. KANZOW, C., and FUKUSHIMA, M., Equivalence of the Generalized Complementarity Problem to Differentiable Unconstrained Minimization, Journal of Optimization Theory and Applications, Vol. 90, pp. 581–603, 1996.

    Google Scholar 

  13. PANG, J. S., A Posteriori Error Bounds for the Linearly-Constrained Variational Inequality Problem, Mathematics of Operations Research, Vol. 12, pp. 474–484, 1987.

    Google Scholar 

  14. LUO, Z. Q., and TSENG, P., A New Class of Merit Functions for the Nonlinear Complementarity Problem, Complementarity and Variational Problems, Edited by M. C. Ferris and J. S. Pang, SIAM (to appear).

  15. COTTLE, R. W., PANG, J. S., and STONE, R. E., The Linear Complementarity Problem, Academic Press, New York, New York, 1992.

    Google Scholar 

  16. MORÉ, J. J., and RHEINBOLDT, W. C., On P-and S-Functions and Related Classes of n-Dimensional Nonlinear Mappings, Linear Algebra and Its Applications, Vol. 6, pp. 45–68, 1973.

    Google Scholar 

  17. YAMASHITA, N., and FUKUSHIMA, M., On Stationary Points of the Implicit Lagrangian for Nonlinear Complementarity Problems, Journal of Optimization Theory and Applications, Vol. 84, pp. 653–663, 1995.

    Google Scholar 

  18. CHEN, B., and HARKER, P. T., Smooth Approximations to Nonlinear Complementarity Problems, SIAM Journal on Optimization (to appear).

  19. DE LUCA, A., FACCHINEI, F., and KANZOW, C., A Semismooth Equation Approach to the Solution of Nonlinear Complementarity Problems, Mathematical Programming, Vol. 75, pp. 407–439, 1996.

    Google Scholar 

  20. FACCHINEI, F., and KANZOW, C., On Unconstrained and Constrained Stationary Points of the Implicit Lagrangian, Journal of Optimization Theory and Applications (to appear).

  21. JIANG, H., Unconstrained Minimization Approaches to Nonlinear Complementarity Problems, Journal of Global Optimization (to appear).

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, University of Hamburg, Hamburg, Germany

    C. Kanzow (Wissenschaftlicher Assistant)

  2. Department of Applied Mathematics and Physics, Graduate School of Engineering, Kyoto University, Kyoto, Japan

    N. Yamashita (Research Fellow of the Japan Society for the Promotion of Science)

  3. Department of Applied Mathematics and Physics, Graduate School of Engineering, Kyoto University, Kyoto, Japan

    M. Fukushima (Professor)

Authors
  1. C. Kanzow
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. Yamashita
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Fukushima
    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

Kanzow, C., Yamashita, N. & Fukushima, M. New NCP-Functions and Their Properties. Journal of Optimization Theory and Applications 94, 115–135 (1997). https://doi.org/10.1023/A:1022659603268

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1022659603268

Search

Navigation