This paper addresses an inconsistency in various definitions of supported non-dominated points within multi-objective combinatorial problems (MOCO). MOCO problems are known to contain supported and unsupported non-dominated points, with the latter typically outnumbering the former. Supported points are, in general, easier to determine, can serve as representations, and are used in two-phase methods to generate the entire non-dominated point set. Despite their importance, several different characterizations for supported efficient solutions (and supported non-dominated points) are used in the literature. While these definitions are equivalent for multi-objective linear problems, they can yield different sets of supported non-dominated points for MOCO problems. We show by an example that these definitions are not equivalent for MOCO or general multi-objective optimization problems. Moreover, we analyze the structural and computational properties of the resulting sets of supported non-dominated points. These considerations motivate us to summarize equivalent definitions and characterizations for supported efficient solutions and to introduce a distinction between supported and weakly supported efficient solutions.

翻译：本文探讨了多目标组合问题中各种关于支撑非支配点定义的不一致性问题。多目标组合问题已知包含支撑与非支撑两类非支配点，且后者数量通常多于前者。支撑点通常更易确定，可作为问题表征，并用于两阶段方法中以生成完整的非支配点集。尽管支撑点具有重要性，现有文献对支撑有效解（及支撑非支配点）存在多种不同定义。这些定义在多目标线性问题中等价，但在多目标组合问题中可能导致不同的支撑非支配点集合。我们通过实例证明这些定义对于多目标组合问题或一般多目标优化问题并不等价。此外，我们分析了由此产生的支撑非支配点集合的结构与计算性质。基于这些分析，我们系统总结了支撑有效解的等价定义与特征，并提出了支撑有效解与弱支撑有效解的区分标准。