Skip to main content
SpringerLink
Log in

Strict Feasibility Conditions in Nonlinear Complementarity Problems

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

Abstract

Strict feasibility plays an important role in the development of the theoryand algorithms of complementarity problems. In this paper, we establishsufficient conditions to ensure strict feasibility of a nonlinearcomplementarity problem. Our analysis method, based on a newly introducedconcept of μ-exceptional sequence, can be viewed as a unified approachfor proving the existence of a strictly feasible point. Some equivalentconditions of strict feasibility are also developed for certaincomplementarity problems. In particular, we show that aP*-complementarity problem is strictly feasible if and only ifits solution set is nonempty and bounded.

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. Kojima, M., Megiddo, N., Noma, T., and Yoshise, A., A Unified Approach to Interior-Point Algorithms for Linear Complementarity Problems, Lecture Note in Computer Sciences, Springer Verlag, New York, NY, Vol. 538, 1991.

    Google Scholar 

  2. Kojima, M., Mizuno, S., and Yoshise, A., A Polynomial Time Algorithm for a Class of Linear Complementarity Problems, Mathematical Programming, Vol. 44, pp. 1–26, 1989.

    Google Scholar 

  3. Kojima, M., Megiddo, N., and Noma, T., Homotopy Continuation Methods forNonlinearComplementar ity Problems, Mathematics of Operations Research, Vol. 16, pp. 754–774, 1991.

    Google Scholar 

  4. Kojima, M., Mizuno, S., and Noma, T., A New Continuation Method for Complementarity Problems with Uniform P-Functions, Mathematical Programming, Vol. 43, pp. 107–113, 1989.

    Google Scholar 

  5. McLinden, L., The Complementarity Problem for Maximal Monotone Multifunctions, Variational Inequalities and Complementarity Problems, Edited by R. W. Cottle, F. Giannessi, and J. L. Lions, John Wiley and Sons, New York, NY, pp. 251–270, 1980.

    Google Scholar 

  6. Megiddo, N., Pathways to the Optimal Set in Linear Programming, Progress in Mathematical Programming: Interior-Point and Related Methods, Edited by N. Megiddo, Springer Verlag, New York, NY, pp. 131–158, 1989.

    Google Scholar 

  7. GÜler, O., Existence of Interior Points and Interior-Point Paths in Nonlinear Monotone Complementarity Problems, Mathematics of Operations Research, Vol. 18, pp. 128–147, 1993.

    Google Scholar 

  8. Karamardian, S., Complementarity Problems over Cones with Monotone and Pseudomonotone Maps, Journal of Optimization Theory and Applications, Vol. 18, pp. 445–454, 1976.

    Google Scholar 

  9. 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 

  10. Zhao, Y. B., and Isac, G., Quasi-P * -Maps, P( τ, α, β)-Maps, Exceptional Family of Elements, and Complementarity Problems, Journal of Optimization Theory and Applications, Vol. 105, pp. 213–231, 2000.

    Google Scholar 

  11. Chen, B., Chen, X., and Kanzow, C., A Penalized Fischer–Burmeister NCPFunction: Theoretical Investigation and Numerical Results, Report, Department of Management and Systems, Washington State University, Pullman, Washington, 1997.

    Google Scholar 

  12. Ravindran, G., and Gowda, M. S., Regularization of P 0 -Function in Box Variational Inequality Problems, SIAM Journal on Optimization, to appear, 2000.

  13. McLinden, L., Stable Monotone Variational Inequalities, Mathematical Programming, Vol. 48, pp. 303–338, 1990.

    Google Scholar 

  14. Ha, C. D., Stability of the Linear Complementarity Problem at a Solution Point, Mathematical Programming, Vol. 31, pp. 327–338, 1985.

    Google Scholar 

  15. Gowda, M. S., and Pang, J. S., On the Boundedness and Stability of Solutions to the Af.ne Variational Inequality Problems, SIAM Journal on Control and Optimization, Vol. 32, pp. 421–441, 1994.

    Google Scholar 

  16. Facchinei, F., Structural and Stability Properties of P 0 -NonlinearComplementarity Problems, Mathematics of Operations Research, Vol. 23, pp. 735–749, 1998.

    Google Scholar 

  17. Facchinei, F., and Pang, J. S., Total Stability of Variational Inequalities, Report, Dipartimento di Informatica e Sistemistica, Universitá di Roma, Roma, Italy, 1998.

    Google Scholar 

  18. Lloyd, N. G., Degree Theory, Cambridge University Press, Cambridge, England, 1978.

    Google Scholar 

  19. Smith, T. E., A Solution Condition for Complementarity Problems with an Application to Spatial Price Equilibrium, Applied Mathematics and Computation, Vol. 15, pp. 61–69, 1984.

    Google Scholar 

  20. Isac, G., Bulavaski, V., and Kalashnikov, V., Exceptional Families, Topological Degree, and Complementarity Problems, Journal of Global Optimization, Vol. 10, pp. 207–225, 1997.

    Google Scholar 

  21. Isac, G., and Obuchowska, W. T., Functions without Exceptional Families of Elements and Complementarity Problems, Journal of Optimization Theory and Applications, Vol. 99, pp. 147–163, 1998.

    Google Scholar 

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

    Google Scholar 

  23. Zhao, Y. B., and Li, D., Characterizations of a Homotopy Solution Mapping for Nonlinear Complementarity Problems, Report, Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, China, 1998.

    Google Scholar 

  24. Karamardian, S., and Schaible, S., Seven Kinds of Monotone Maps, Journal of Optimization Theory and Applications, Vol. 66, pp. 37–46, 1990.

    Google Scholar 

  25. Zhao, Y. B., and Han, J., Exceptional Family of Elements fora Variational Inequality Problem and Its Applications, Journal of Global Optimization, Vol. 14, pp. 313–330, 1999.

    Google Scholar 

  26. Cottle, R. W., Pang, J. S., and Venkateswaran, V., Sufficient Matrices and the Linear Complementarity Problems, Linear Algebra and Its Applications, Vols. 114/115, pp. 231–249, 1989.

    Google Scholar 

  27. Karamardian, S., The Complementarity Problem, Mathematical Programming, Vol. 2, pp. 107–129, 1972.

    Google Scholar 

  28. MorÉ, J. J., Coercivity Conditions in Nonlinear Complementarity Problems, SIAM Review, Vol. 16, pp. 1–16, 1974.

    Google Scholar 

  29. Zhao, Y. B., Existence of a Solution to a NonlinearVar iational Inequality under Generalized Positive Homogeneity, Operations Research Letters, Vol. 25, pp. 231–239, 1999.

    Google Scholar 

  30. Tseng, T., Growth Behavior of a Class of Merit Functions for the Nonlinear Complementarity Problems, Journal of Optimization Theory and Applications, Vol. 89, pp. 17–37, 1996.

    Google Scholar 

  31. Chen, B., and Harker, P. T., Smooth Approximations to Nonlinear Complementarity Problems, SIAM Journal on Optimization, Vol. 7, pp. 403–420, 1997.

    Google Scholar 

  32. Luo, Z. Q., and Tseng, T., A New Class of Merit Functions for the Nonlinear Complementarity Problems, Complementarity and Variational Problems: State of the Art, Edited by M. C. Ferris and J. S. Pang, SIAM, Philadelphia, Pennsylvania, pp. 204–225, 1997.

    Google Scholar 

  33. Gowda, M. S., and Tawhid, M. A., Existence and Limiting Behaviorof Trajectories Associated with P 0 -Equations, Computational Optimization and Applications, Vol. 12, pp. 229–251, 1999.

    Google Scholar 

  34. Marcotte, P., and Wu, J. H., On the Convergence of Projection Methods: Application to the Decomposition of Af.ne Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 85, pp. 347–362, 1995.

    Google Scholar 

  35. Zhu, D. L., and Marcotte, P., Cocoercivity and Its Role in the Convergence of Iterative Schemes for Solving Variational Inequalities, SIAM Journal on Optimization, Vol. 6, pp. 714–726, 1996.

    Google Scholar 

  36. Robinson, S., Generalized Equations and Their Solutions, Part 1: Basic Theory, Mathematical Programming, Vol. 10, pp. 128–141, 1979.

    Google Scholar 

  37. Isac, G., Exceptional Families of Elements, Feasibility, and Complementarity, Report, Department of Mathematics and Computer Sciences, Royal Military College of Canada, Kingston, Canada, 1998.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of System Engineering and Engineering Management, Chinese University of Hong Kong, Shatin, NT, Hong Kong

    Y. B. Zhao (Researcher)

  2. Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, China

    Y. B. Zhao (Researcher)

  3. Department of System Engineering and Engineering Management, Chinese University of Hong Kong, Shatin, NT, Hong Kong

    D. Li (Associate Professor)

Authors
  1. Y. B. Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Li
    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

Zhao, Y.B., Li, D. Strict Feasibility Conditions in Nonlinear Complementarity Problems. Journal of Optimization Theory and Applications 107, 641–664 (2000). https://doi.org/10.1023/A:1026459501988

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1026459501988

Search

Navigation