Theory of the hypervolume indicator

The hypervolume indicator is a set measure used in evolutionary multiobjective optimization to evaluate the performance of search algorithms and to guide the search. Multiobjective evolutionary algorithms using the hypervolume indicator transform multiobjective problems into single objective ones by searching for a finite set of solutions maximizing the corresponding hypervolume indicator. In this paper, we theoretically investigate how those optimal μ--distributions-finite sets of μ solutions maximizing the hypervolume indicator-are spread over the Pareto front of biobjective problems. This problem is of high importance for practical applications as these sets characterize the preferences that the hypervolume indicator encodes, i.e., which types of Pareto set approximations are favored.