Consistencies and contradictions of performance metrics in multiobjective optimization.

An important consideration of multiobjective optimization (MOO) is the quantitative metrics used for defining the optimality of different solution sets, which is also the basic principle for the design and evaluation of MOO algorithms. Although a plethora of performance metrics have been proposed in the MOO context, there has been a lack of insights on the relationships between metrics. In this paper, we first group the major MOO metrics proposed to date according to four core performance criteria considered in the literature, namely, capacity, convergence, diversity, and convergence-diversity. Then, a comprehensive study is conducted to investigate the relationships among representative group metrics, including generational distance, ϵ-indicator (I(1)ϵ+), spread (∆), generalized spread (∆∗), inverted generational distance, and hypervolume. Experimental results indicated that these six metrics show high consistencies when Pareto fronts (PFs) are convex, whereas they show certain contradictions on concave PFs.