Optimal µ-Distributions for the Hypervolume Indicator for Problems with Linear Bi-objective Fronts: Exact and Exhaustive Results

To simultaneously optimize multiple objective functions, several evolutionary multiobjective optimization (EMO) algorithms have been proposed. Nowadays, often set quality indicators are used when comparing the performance of those algorithms or when selecting "good" solutions during the algorithm run. Hence, characterizing the solution sets that maximize a certain indicator is crucial--complying with the optimization goal of many indicator-based EMO algorithms. If these optimal solution sets are upper bounded in size, e.g., by the population size µ, we call them optimal µ-distributions. Recently, optimal µ-distributions for the well-known hypervolume indicator have been theoretically analyzed, in particular, for bi-objective problems with a linear Pareto front. Although the exact optimal µ-distributions have been characterized in this case, not all possible choices of the hypervolume's reference point have been investigated. In this paper, we revisit the previous results and rigorously characterize the optimal µ-distributions also for all other reference point choices. In this sense, our characterization is now exhaustive as the result holds for any linear Pareto front and for any choice of the reference point and the optimal µ-distributions turn out to be always unique in those cases. We also prove a tight lower bound (depending on µ) such that choosing the reference point above this bound ensures the extremes of the Pareto front to be always included in optimal µ-distributions.