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.
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Acknowledgements
The authors are grateful to the anonymous referees for their valuable comments and suggestions, which help to improve the paper. This research was supported by the National Natural Science Foundation of China (Grants: 11671329, 11601437, 71942006, 71671142) and the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant: KJ1713334).
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Communicated by Guang-Ya Chen.
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Pan, S., Lu, S., Meng, K. et al. Trade-Off Ratio Functions for Linear and Piecewise Linear Multi-objective Optimization Problems. J Optim Theory Appl 188, 402–419 (2021). https://doi.org/10.1007/s10957-020-01788-6
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DOI: https://doi.org/10.1007/s10957-020-01788-6
Keywords
- Multi-objective optimization
- Linear and piecewise linear programming
- Trade-off ratio function
- Proper efficiency