Exact Calculation and Properties of the R2 Multiobjective Quality Indicator

Quality indicators play an essential role in evolutionary multiobjective optimization (EMO). Most likely the most often used quality indicator in EMO is hypervolume, due to its strict monotonicity with respect to the dominance relation. However, hypervolume is not free of some weak points. For example, a number of recent papers pointed out its high sensitivity to the specification of the reference point. Furthermore, hypervolume is based on fully geometric reasoning which may lead to some undesired results. Thus, it is worth to consider also other quality indicators. In this article, we prove that another well-known $R2$ quality indicator is also strictly monotonic with respect to the dominance relation when calculated exactly and the reference point strongly dominates any solution in the evaluated set. Furthermore, we adapt the improved quick hypervolume algorithm to the exact calculation of $R2$ indicator. To our knowledge, this is the first exact algorithm for $R2$ calculation with publicly available implementation. In addition, through both theoretical analysis and computational experiments, we show that $R2$ performs consistently for Pareto fronts with different shapes. We discuss also differences of Pareto fronts representations generated by an indicator-based EMO with hypervolume and $R2$ , where the latter tends to generate solutions having a high chance to be preferred by the decision maker, not necessarily uniformly distributed in geometric sense. All of these results make $R2$ a sound alternative or a complement to hypervolume in EMO.