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Adaptive Smoothing Method, Deterministically Computable Generalized Jacobians, and the Newton Method

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  • H. Xu

Abstract

In this note, we show that a well-known integral method, which was used by Mayne and Polak to compute an ∈-subgradient, can be exploited to compute deterministically an element of the plenary hull of the Clarke generalized Jacobian of a locally Lipschitz mapping regardless of its structure. In particular, we show that, when a locally Lipschitz mapping is piecewise smooth, we are able to compute deterministically an element of the Clarke generalized Jacobian by the adaptive smoothing method. Consequently, we show that the Newton method based on the plenary hull of the Clarke generalized Jacobian can be implemented in a deterministic way for solving Lipschitz nonsmooth equations.

Suggested Citation

  • H. Xu, 2001. "Adaptive Smoothing Method, Deterministically Computable Generalized Jacobians, and the Newton Method," Journal of Optimization Theory and Applications, Springer, vol. 109(1), pages 215-224, April.
    • Handle: RePEc:spr:joptap:v:109:y:2001:i:1:d:10.1023_a:1017526207997
    DOI: 10.1023/A:1017526207997
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    References listed on IDEAS

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    1. H. Xu & X. W. Chang, 1997. "Approximate Newton Methods for Nonsmooth Equations," Journal of Optimization Theory and Applications, Springer, vol. 93(2), pages 373-394, May.
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    Cited by:

    1. 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.
    2. D. Ralph & H. Xu, 2005. "Implicit Smoothing and Its Application to Optimization with Piecewise Smooth Equality Constraints1," Journal of Optimization Theory and Applications, Springer, vol. 124(3), pages 673-699, March.

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