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

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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.

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Acknowledgements

The authors would like to thank the anonymous referees and the editor for their helpful comments and suggestions which have led to the improvement of the early version of this paper. This work was supported by the National Science Foundation of China (11001187, 108311009) and the Research Grants Council of Hong Kong (PolyU5306/11E).

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Authors and Affiliations

  1. Department of Mathematics, Sichuan University, Chengdu, Sichuan, P.R. China

    Ya Ping Fang & Nan Jing Huang

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hum, Kowloon, Hong Kong

    Xiao Qi Yang

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  1. Ya Ping Fang
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  2. Nan Jing Huang
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  3. Xiao Qi Yang
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Correspondence to Ya Ping Fang.

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Fang, Y.P., Huang, N.J. & Yang, X.Q. Local Smooth Representations of Parametric Semiclosed Polyhedra with Applications to Sensitivity in Piecewise Linear Programs. J Optim Theory Appl 155, 810–839 (2012). https://doi.org/10.1007/s10957-012-0089-3

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  • DOI: https://doi.org/10.1007/s10957-012-0089-3

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