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On Minimizing Some Merit Functions for Nonlinear Complementarity Problems under H-Differentiability

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  • M. A. Tawhid

    (Thompson Rivers University)

  • J. L. Goffin

    (McGill University)

Abstract

In this paper, we describe the H-differentials of some well known NCP functions and their merit functions. We show how, under appropriate conditions on an H-differential of f, minimizing a merit function corresponding to f leads to a solution of the nonlinear complementarity problem. Our results give a unified treatment of such results for C 1-functions, semismooth-functions, and locally Lipschitzian functions. Illustrations are given to show the usefulness of our results. We present also a result on the global convergence of a derivative-free descent algorithm for solving the nonlinear complementarity problem.

Suggested Citation

  • M. A. Tawhid & J. L. Goffin, 2008. "On Minimizing Some Merit Functions for Nonlinear Complementarity Problems under H-Differentiability," Journal of Optimization Theory and Applications, Springer, vol. 139(1), pages 127-140, October.
    • Handle: RePEc:spr:joptap:v:139:y:2008:i:1:d:10.1007_s10957-008-9409-z
    DOI: 10.1007/s10957-008-9409-z
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    References listed on IDEAS

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    1. Liqun Qi, 1993. "Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 227-244, February.
    2. Yoon Song & M. Seetharama Gowda & G. Ravindran, 2000. "On Characterizations of P - and P 0 -Properties in Nonsmooth Functions," Mathematics of Operations Research, INFORMS, vol. 25(3), pages 400-408, August.
    3. A. Fischer, 1998. "New Constrained Optimization Reformulation of Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 97(1), pages 105-117, April.
    4. A. Fischer & V. Jeyakumar & D. T. Luc, 2001. "Solution Point Characterizations and Convergence Analysis of a Descent Algorithm for Nonsmooth Continuous Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 110(3), pages 493-513, September.
    5. F. Facchinei & C. Kanzow, 1997. "On Unconstrained and Constrained Stationary Points of the Implicit Lagrangian," Journal of Optimization Theory and Applications, Springer, vol. 92(1), pages 99-115, January.
    6. M.A. Tawhid, 2002. "On the Local Uniqueness of Solutions of Variational Inequalities Under H-Differentiability," Journal of Optimization Theory and Applications, Springer, vol. 113(1), pages 149-164, April.
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