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Journal of Optimization Theory and Applications

, Volume 110, Issue 3, pp 493–513 | Cite as

Solution Point Characterizations and Convergence Analysis of a Descent Algorithm for Nonsmooth Continuous Complementarity Problems

  • A. Fischer
  • V. Jeyakumar
  • D. T. Luc
Article

Abstract

We consider a nonlinear complementarity problem defined by a continuous but not necessarily locally Lipschitzian map. In particular, we examine the connection between solutions of the problem and stationary points of the associated Fischer–Burmeister merit function. This is done by deriving a new necessary optimality condition and a chain rule formula for composite functions involving continuous maps. These results are given in terms of approximate Jacobians which provide the foundation for analyzing continuous nonsmooth maps. We also prove a result on the global convergence of a derivative-free descent algorithm for solving the complementarity problem. To this end, a concept of directional monotonicity for continuous maps is introduced.

Approximate Jacobians nonsmooth continuous maps complementarity problems nonsmooth analysis descent algorithms 

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References

  1. 1.
    Ferris, M. C., and Pang, J. S., Engineering and Economic Applications of Complementary Problems, SIAM Review, Vol. 39, pp. 669–713, 1997.Google Scholar
  2. 2.
    Tin-Loi, F., and Ferris, M. C., Complementarity Problems in Engineering Mechanics: Models and Solution, Computational Mechanics for the Next Millennium, Edited by C. M. Wang, K. H. Lee, and K. K. Ang, Elsevier Science, Amsterdam, Holland, pp. 1029–1036, 1999.Google Scholar
  3. 3.
    Pang, J. S., Complementarity Problems, Handbook of Global Optimization, Edited by R. Horst and P. M. Pardalos, Kluwer Academic Publishers, Dordrecht, Holland, pp. 271–338, 1994.Google Scholar
  4. 4.
    Ferris, M. C., and Pang, J. S., Complementarity and Variational Problems: State of the Art, SIAM, Philadelphia, Pennsylvania, 1997.Google Scholar
  5. 5.
    Fischer, A., Solution of Monotone Complementarity Problems with Locally Lipschitzian Functions, Mathematical Programming, Vol. 76, pp. 513–532, 1997.Google Scholar
  6. 6.
    Jiang, H., Unconstrained Minimization Approaches to Nonlinear Complementarity Problems, Journal of Global Optimization, Vol. 9, pp. 169–181, 1996.Google Scholar
  7. 7.
    Jeyakumar, V., and Luc, D. T., Approximate Jacobian Matrices for Nonsmooth Continuous Maps and C 1-Optimization, SIAM Journal on Control and Optimization, Vol. 36, pp. 1815–1832, 1998.Google Scholar
  8. 8.
    Jeyakumar, V., Luc, D. T., and Schaible, S., Characterizations of Generalized Monotone Nonsmooth Continuous Maps Using Approximate Jacobians, Journal of Convex Analysis, Vol. 5, pp. 119–132, 1998.Google Scholar
  9. 9.
    Jeyakumar, V., and Wang, X., Approximate Hessian Matrices and Second-Order Optimality Conditions for Nonlinear Programming Problems with C 1 -Data, Journal of the Australian Mathematical Society, Vol. 40B, pp. 403–420, 1999.Google Scholar
  10. 10.
    Wang, X., and Jeyakumar, V., A Sharp Lagrange Multiplier Rule for Nonsmooth Mathematical Programming Problems Involving Equality Constraints, SIAM Journal on Optimization, Vol. 10, pp. 1136–1148, 2000.Google Scholar
  11. 11.
    Geiger, C., and Kanzow, C., On the Resolution of Monotone Complementarity Problems, Computational Optimization and Applications, Vol. 5, pp. 155–173, 1996.Google Scholar
  12. 12.
    Burke, J. V., and Overton, M. L., Variational Analysis of Non-Lipschitz Spectral Functions, Mathematical Programming, Vol. 39, pp. 317–351, 2001.Google Scholar
  13. 13.
    Fischer, A., A Special Newton-Type Optimization Method, Optimization, Vol. 24, pp. 269–284, 1992.Google Scholar
  14. 14.
    Facchinei, F., and Soares, J., A New Merit Function for Nonlinear Complementarity Problems and a Related Algorithm, SIAM Journal on Optimization, Vol. 7, pp. 225–247, 1997.Google Scholar
  15. 15.
    Tseng, P., Further Applications of a Splitting Algorithm to Decomposition in Variational Inequalities and Convex Programming, Mathematical Programming, Vol. 48, pp. 249–264, 1990.Google Scholar
  16. 16.
    Demyanov, V. F., and Rubinov, A. M., Constructive Nonsmooth Analysis, Verlag Peter Lang, Frankfurt am Main, Germany, 1995.Google Scholar
  17. 17.
    Rockafellar, R. T., and Wets, R., Variational Analysis, Springer Verlag, Berlin, Germany, 1998.Google Scholar
  18. 18.
    Jeyakumar, V., and Luc, D. T., Open Mapping Theorem Using Unbounded Generalized Jacobians, Applied Mathematics Report AMR99/20, University of New South Wales, Sydney, NSW, Australia, 1999, Nonlinear Analysis, 2001 (to appear).Google Scholar
  19. 19.
    Jeyakumar, V., Luc, D. T., and Wang, X., Lagrange Multipliers for Equality Constraints without Lipschitz Continuity, Applied Mathematics Report AMR00/1, University of New South Wales, Sydney, NSW, Australia, 2000.Google Scholar
  20. 20.
    Luc, D. T., Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, Germany, Vol. 319, 1989.Google Scholar
  21. 21.
    Luc, D. T., Recession Maps and Applications, Optimization, Vol. 27, pp. 1–15, 1993.Google Scholar
  22. 22.
    Jeyakumar, V., and Luc, D. T., Nonsmooth Calculus, Minimality, and Monotonicity of Convexificators, Journal of Optimization Theory and Applications, Vol. 101, pp. 599–621, 1999.Google Scholar
  23. 23.
    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
  24. 24.
    Chen, B., Chen, X., and Kanzow, C., A Penalized FischerBurmeister NCP Function, Mathematical Programming, Vol. 88, pp. 211–216, 2000.Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • A. Fischer
  • V. Jeyakumar
  • D. T. Luc

There are no affiliations available

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