On Supportedness in Multi-Objective Combinatorial Optimization

TL;DR

This paper investigates the inconsistencies in defining supported non-dominated points in multi-objective combinatorial optimization, clarifies their properties, and proposes a unified framework distinguishing supported and weakly supported solutions.

Contribution

It identifies inconsistencies in existing definitions for supported points in MOCO, analyzes their properties, and proposes a unified characterization and distinction between supported and weakly supported solutions.

Findings

Different definitions yield different supported point sets in MOCO.

Supported points are structurally and computationally significant.

A unified framework for supported and weakly supported solutions is proposed.

Abstract

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.