Transformation-based Hypervolume Indicator: A Framework for Designing Hypervolume Variants

The hypervolume indicator is a popular performance indicator in the field of Evolutionary Multi-objective optimization (EMO). However, there are two issues associated with it in addition to its large calculation cost for many-objective problems. The first issue is that the maximization of the hypervolume indicator leads to a non-uniform solution set on a nonlinear Pareto front. The second issue is that it cannot handle preference information. To address these two issues, some hypervolume variants have been proposed in the literature. In this paper, first we review these variants and extract the common characteristic among them, i.e., all these variants can be converted to the standard hypervolume indicator with a transformed solution set. Based on this observation, we propose the transformation-based hypervolume indicator, which is a framework for designing hypervolume variants. Then, we propose two new hypervolume variants based on our framework. Empirical studies suggest the effectiveness of the proposed variants for addressing the abovementioned two issues. Our experimental results also suggest the possibility of designing other hypervolume variants for different purposes using our framework.