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.

中文翻译：

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线性和分段线性多目标优化问题的权衡比函数

在本文中，我们引入了与著名的 Geoffrion 对多目标优化问题的适当效率密切相关的权衡比函数的概念，并研究了其有界性。对于线性多目标优化问题，我们证明了权衡比函数在有效解集上是有界的。对于分段线性多目标优化问题，我们证明所有有效的解决方案在 Borwein 的意义上总是适当有效的，而且，当且仅当衰退条件成立时，所有有效的解决方案在 Geoffrion 的意义上都是适当的。最后，我们提供了一个例子来说明权衡比函数在分段线性多目标优化问题的有效解集上可能是无界的，即使衰退条件成立，