The Hypervolume Indicator Revisited: On the Design of Pareto-compliant Indicators Via Weighted Integration

The design of quality measures for approximations of the Pareto-optimal set is of high importance not only for the performance assessment, but also for the construction of multiobjective optimizers. Various measures have been proposed in the literature with the intention to capture different preferences of the decision maker. A quality measure that possesses a highly desirable feature is the hypervolume measure: whenever one approximation completely dominates another approximation, the hypervolume of the former will be greater than the hypervolume of the latter. Unfortunately, this measure—as any measure inducing a total order on the search space—is biased, in particular towards convex, inner portions of the objective space. Thus, an open question in this context is whether it can be modified such that other preferences such as a bias towards extreme solutions can be obtained. This paper proposes a methodology for quality measure design based on the hypervolume measure and demonstrates its usefulness for three types of preferences.