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

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

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.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:110:y:2001:i:3:d:10.1023_a:1017580126509
    DOI: 10.1023/A:1017580126509
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    References listed on IDEAS

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    1. V. Jeyakumar & D. T. Luc, 1999. "Nonsmooth Calculus, Minimality, and Monotonicity of Convexificators," Journal of Optimization Theory and Applications, Springer, vol. 101(3), pages 599-621, June.
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    Cited by:

    1. V. Jeyakumar & G. M. Lee & N. Dinh, 2004. "Lagrange Multiplier Conditions Characterizing the Optimal Solution Sets of Cone-Constrained Convex Programs," Journal of Optimization Theory and Applications, Springer, vol. 123(1), pages 83-103, October.
    2. 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.
    3. Pin-Bo Chen & Peng Zhang & Xide Zhu & Gui-Hua Lin, 2020. "Modified Jacobian smoothing method for nonsmooth complementarity problems," Computational Optimization and Applications, Springer, vol. 75(1), pages 207-235, January.

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