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Implicit Smoothing and Its Application to Optimization with Piecewise Smooth Equality Constraints1

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  • D. Ralph

    (University of Cambridge)

  • H. Xu

    (University of Southampton)

Abstract

In this paper, we discuss the smoothing of an implicit function defined by a nonsmooth underdetermined system of equations F(y,z) = 0. We apply a class of parametrized smoothing methods to smooth F and investigate the limiting behavior of the implicit function solving the smoothed equations. In particular, we discuss the approximation of the Clarke generalized Jacobian of the implicit function when F is piecewise smooth. As an application, we present an analysis of the generalized Karush-Kuhn-Tucker conditions of different forms for a piecewise-smooth equality-constrained minimization problem.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:124:y:2005:i:3:d:10.1007_s10957-004-1180-1
    DOI: 10.1007/s10957-004-1180-1
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    References listed on IDEAS

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    1. Jong-Shi Pang & Daniel Ralph, 1996. "Piecewise Smoothness, Local Invertibility, and Parametric Analysis of Normal Maps," Mathematics of Operations Research, INFORMS, vol. 21(2), pages 401-426, May.
    2. 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.
    3. Liqun Qi, 1993. "Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 227-244, February.
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    Cited by:

    1. Huifu Xu & Fanwen Meng, 2007. "Convergence Analysis of Sample Average Approximation Methods for a Class of Stochastic Mathematical Programs with Equality Constraints," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 648-668, August.
    2. Gui-Hua Lin & Mei-Ju Luo & Jin Zhang, 2016. "Smoothing and SAA method for stochastic programming problems with non-smooth objective and constraints," Journal of Global Optimization, Springer, vol. 66(3), pages 487-510, November.
    3. Huifu Xu & Dali Zhang, 2013. "Stochastic Nash equilibrium problems: sample average approximation and applications," Computational Optimization and Applications, Springer, vol. 55(3), pages 597-645, July.
    4. Victor DeMiguel & Huifu Xu, 2009. "A Stochastic Multiple-Leader Stackelberg Model: Analysis, Computation, and Application," Operations Research, INFORMS, vol. 57(5), pages 1220-1235, October.
    5. 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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