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Local Smooth Representations of Parametric Semiclosed Polyhedra with Applications to Sensitivity in Piecewise Linear Programs

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  • Ya Ping Fang

    (Sichuan University)

  • Nan Jing Huang

    (Sichuan University)

  • Xiao Qi Yang

    (The Hong Kong Polytechnic University)

Abstract

In this paper, we establish the equivalence between the half-space representation and the vertex representation of a smooth parametric semiclosed polyhedron. By virtue of the smooth representation result, we prove that the solution set of a smooth parametric piecewise linear program can be locally represented as a finite union of parametric semiclosed polyhedra generated by finite smooth functions. As consequences, we prove that the corresponding marginal function is differentiable and the solution map admits a differentiable selection.

Suggested Citation

  • Ya Ping Fang & Nan Jing Huang & Xiao Qi Yang, 2012. "Local Smooth Representations of Parametric Semiclosed Polyhedra with Applications to Sensitivity in Piecewise Linear Programs," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 810-839, December.
    • Handle: RePEc:spr:joptap:v:155:y:2012:i:3:d:10.1007_s10957-012-0089-3
    DOI: 10.1007/s10957-012-0089-3
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    References listed on IDEAS

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    1. Julia L. Higle & Stein W. Wallace, 2003. "Sensitivity Analysis and Uncertainty in Linear Programming," Interfaces, INFORMS, vol. 33(4), pages 53-60, August.
    2. Darinka Dentcheva & Andrzej Ruszczynski, 2004. "Optimization Under First Order Stochastic Dominance Constraints," GE, Growth, Math methods 0403002, University Library of Munich, Germany, revised 07 Aug 2005.
    3. Martin R. Young, 1998. "A Minimax Portfolio Selection Rule with Linear Programming Solution," Management Science, INFORMS, vol. 44(5), pages 673-683, May.
    4. Hadigheh, Alireza Ghaffari & Terlaky, Tamas, 2006. "Sensitivity analysis in linear optimization: Invariant support set intervals," European Journal of Operational Research, Elsevier, vol. 169(3), pages 1158-1175, March.
    5. X. Q. Yang & N. D. Yen, 2010. "Structure and Weak Sharp Minimum of the Pareto Solution Set for Piecewise Linear Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 113-124, October.
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    Cited by:

    1. Margarita M. L. Rodríguez & José Vicente-Pérez, 2017. "On Finite Linear Systems Containing Strict Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 131-154, April.

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