Theory of the Hypervolume Indicator: Optimal µ-Distributions and the Choice of the Reference Point

The hypervolume indicator is a set measure used in evolu-tionary multiobjective optimization to evaluate the perfor-mance of search algorithms and to guide the search. Mul-tiobjective evolutionary algorithms using the hypervolume indicator transform multiobjective problems into single ob-jective ones by searching for a finite set of solutions max-imizing the corresponding hypervolume indicator. In this paper, we theoretically investigate how those optimal µ-distributions—finite sets of µ solutions maximizing the hy-pervolume indicator—are spread over the Pareto front of biobjective problems. This problem is of high importance for practical applications as these sets characterize the pref-erences that the hypervolume indicator encodes, i.e., which types of Pareto set approximations are favored