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

• Published in: Journal of optimization theory and applications Vol. 188; no. 2; pp. 402 - 419
• Main Authors: Pan, Siming, Lu, Shaokai, Meng, Kaiwen, Zhu, Shengkun
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2021; Springer Nature B.V
• Subjects: Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Engineering; Mathematics; Mathematics and Statistics; Multiple objective analysis; Operations Research/Decision Theory; Optimization; Recession; Theory of Computation; Tradeoffs; Trade-off ratio function; Linear and piecewise linear programming; 90C29; Multi-objective optimization; 90C05; Proper efficiency; 46N10
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-020-01788-6

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.