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

Siming Pan (), Shaokai Lu (), Kaiwen Meng () and Shengkun Zhu ()
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Siming Pan: Southwest Jiaotong University
Shaokai Lu: Southwest Jiaotong University
Kaiwen Meng: Southwestern University of Finance and Economics
Shengkun Zhu: Southwestern University of Finance and Economics

Journal of Optimization Theory and Applications, 2021, vol. 188, issue 2, No 5, 402-419

Abstract: 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.

Keywords: Multi-objective optimization; Linear and piecewise linear programming; Trade-off ratio function; Proper efficiency; 46N10; 90C05; 90C29 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10957-020-01788-6

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