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Exact Penalty and Optimality Condition for Nonseparable Continuous Piecewise Linear Programming

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  • Xiaolin Huang

    (Tsinghua University)

  • Jun Xu

    (Tsinghua University)

  • Shuning Wang

    (Tsinghua University)

Abstract

Utilizing compact representations for continuous piecewise linear functions, this paper discusses some theoretical properties for nonseparable continuous piecewise linear programming. The existence of exact penalty for continuous piecewise linear programming is proved, which allows us to concentrate on unconstrained problems. For unconstrained problems, we give a sufficient and necessary local optimality condition, which is based on a model with universal representation capability and hence applicable to arbitrary continuous piecewise linear programming. From the gained optimality condition, an algorithm is proposed and evaluated by numerical experiments, where the theoretical properties are illustrated as well.

Suggested Citation

  • Xiaolin Huang & Jun Xu & Shuning Wang, 2012. "Exact Penalty and Optimality Condition for Nonseparable Continuous Piecewise Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 145-164, October.
    • Handle: RePEc:spr:joptap:v:155:y:2012:i:1:d:10.1007_s10957-012-0032-7
    DOI: 10.1007/s10957-012-0032-7
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    References listed on IDEAS

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    1. Keely L. Croxton & Bernard Gendron & Thomas L. Magnanti, 2003. "Models and Methods for Merge-in-Transit Operations," Transportation Science, INFORMS, vol. 37(1), pages 1-22, February.
    2. Keely L. Croxton & Bernard Gendron & Thomas L. Magnanti, 2003. "A Comparison of Mixed-Integer Programming Models for Nonconvex Piecewise Linear Cost Minimization Problems," Management Science, INFORMS, vol. 49(9), pages 1268-1273, September.
    3. Juan Pablo Vielma & Shabbir Ahmed & George Nemhauser, 2010. "Mixed-Integer Models for Nonseparable Piecewise-Linear Optimization: Unifying Framework and Extensions," Operations Research, INFORMS, vol. 58(2), pages 303-315, April.
    4. E. M. L. Beale & P. J. Coen & A. D. J. Flowerdew, 1965. "Separable Programming Applied to an Ore Purchasing Problem," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 14(2-3), pages 89-101, November.
    5. Ahmet B. Keha & Ismael R. de Farias & George L. Nemhauser, 2006. "A Branch-and-Cut Algorithm Without Binary Variables for Nonconvex Piecewise Linear Optimization," Operations Research, INFORMS, vol. 54(5), pages 847-858, October.
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