An Empirical Assessment of the Properties of Inverted Generational Distance on Multi- and Many-Objective Optimization

Abstract

The inverted generational distance (IGD) is a metric for assessing the quality of approximations to the Pareto front obtained by multi-objective optimization algorithms. The IGD has become the most commonly used metric in the context of many-objective problems, i.e., those with more than three objectives. The averaged Hausdorff distance and $\textit{IGD}^+$ are variants of the IGD proposed in order to overcome its major drawbacks. In particular, the IGD is not Pareto compliant and its conclusions may strongly change depending on the size of the reference front. It is also well-known that different metrics assign more importance to various desired features of approximation fronts, and thus, they may disagree when ranking them. However, the precise behavior of the IGD variants is not well-understood yet. In particular, $\textit{IGD}^+$, the only IGD variant that is weakly Pareto-compliant, has received significantly less attention. This paper presents an empirical analysis of the IGD variants. Our experiments evaluate how these metrics are affected by the most important factors that intuitively describe the quality of approximation fronts, namely, spread, distribution and convergence. The results presented here already reveal interesting insights. For example, we conclude that, in order to achieve small IGD or $\textit{IGD}^+$ values, the approximation front size should match the reference front size.