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Trade-Off Ratio Functions for Linear and Piecewise Linear Multi-objective Optimization Problems

Author

Listed:
  • Siming Pan

    (Southwest Jiaotong University
    Officers College of People Armed Police)

  • Shaokai Lu

    (Southwest Jiaotong University)

  • Kaiwen Meng

    (Southwestern University of Finance and Economics)

  • Shengkun Zhu

    (Southwestern University of Finance and Economics)

Abstract

In this paper, we introduce the concept of trade-off ratio function, which is closely related to the well-known Geoffrion’s proper efficiency for multi-objective optimization problems, and investigate its boundedness property. For linear multi-objective optimization problems, we show that the trade-off ratio function is bounded on the efficient solution set. For piecewise linear multi-objective optimization problems, we show that all efficient solutions are always properly efficient in Borwein’s sense, and moreover, all efficient solutions are properly efficient in Geoffrion’s sense if and only if a recession condition holds. Finally, we provide an example to illustrate that the trade-off ratio function may be unbounded on the efficient solution set to piecewise linear multi-objective optimization problems, even if the recession condition holds, while it is bounded on the supported efficient solution set.

Suggested Citation

  • Siming Pan & Shaokai Lu & Kaiwen Meng & Shengkun Zhu, 2021. "Trade-Off Ratio Functions for Linear and Piecewise Linear Multi-objective Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 188(2), pages 402-419, February.
    • Handle: RePEc:spr:joptap:v:188:y:2021:i:2:d:10.1007_s10957-020-01788-6
    DOI: 10.1007/s10957-020-01788-6
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    References listed on IDEAS

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    1. Dinh The Luc, 2016. "Multiobjective Linear Programming," Springer Books, Springer, edition 1, number 978-3-319-21091-9, October.
    2. Ya Ping Fang & Kaiwen Meng & Xiao Qi Yang, 2012. "Piecewise Linear Multicriteria Programs: The Continuous Case and Its Discontinuous Generalization," Operations Research, INFORMS, vol. 60(2), pages 398-409, April.
    3. H. Isermann, 1974. "Technical Note—Proper Efficiency and the Linear Vector Maximum Problem," Operations Research, INFORMS, vol. 22(1), pages 189-191, February.
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