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On Optimization Over the Efficient Set of a Multiple Objective Linear Programming Problem

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  • Erjiang Sun

    (Southern California Edison)

Abstract

In this paper, we provide a mixed-integer programming approach for solving the problem of minimizing a real-valued function over the efficient set of a multiple objective linear program problem. Instead of solving the problem directly, we introduce a new problem of minimizing the objective function subject to some linear constraints with additional binary variables. We show under certain conditions that the two problems are equivalent. When the objective function of the original problem is a linear or convex function, the new problem is a linear or convex programming problem, respectively, with some binary variables. These problems can be solved as mixed-integer programs with current state-of-art mixed-integer programming solvers.

Suggested Citation

  • Erjiang Sun, 2017. "On Optimization Over the Efficient Set of a Multiple Objective Linear Programming Problem," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 236-246, January.
    • Handle: RePEc:spr:joptap:v:172:y:2017:i:1:d:10.1007_s10957-016-1030-y
    DOI: 10.1007/s10957-016-1030-y
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    References listed on IDEAS

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    1. Harold Benson, 2012. "An outcome space algorithm for optimization over the weakly efficient set of a multiple objective nonlinear programming problem," Journal of Global Optimization, Springer, vol. 52(3), pages 553-574, March.
    2. Benson, Harold P., 1986. "An algorithm for optimizing over the weakly-efficient set," European Journal of Operational Research, Elsevier, vol. 25(2), pages 192-199, May.
    3. Carla Antoni & Franco Giannessi, 2014. "Some Remarks on Bi-level Vector Extremum Problems," Springer Optimization and Its Applications, in: Vladimir F. Demyanov & Panos M. Pardalos & Mikhail Batsyn (ed.), Constructive Nonsmooth Analysis and Related Topics, edition 127, pages 137-157, Springer.
    4. Horst, Reiner & Thoai, Nguyen V., 1999. "Maximizing a concave function over the efficient or weakly-efficient set," European Journal of Operational Research, Elsevier, vol. 117(2), pages 239-252, September.
    5. H. P. Benson, 2010. "Simplicial Branch-and-Reduce Algorithm for Convex Programs with a Multiplicative Constraint," Journal of Optimization Theory and Applications, Springer, vol. 145(2), pages 213-233, May.
    6. Benson, Harold P. & Sun, Erjiang, 2009. "Branch-and-reduce algorithm for convex programs with additional multiplicative constraints," European Journal of Operational Research, Elsevier, vol. 199(1), pages 1-8, November.
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