Optimal µ-Distributions for the Hypervolume Indicator for Problems With Linear BiObjective Fronts: Exact and Exhaustive Results, in "Simulated Evolution And Learning (SEAL-2010)", Inde Kanpur, November 2010, corrected author version

Abstract. To simultaneously optimize multiple objective functions, several evo-lutionary 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 al-gorithms. 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. Al-though 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 op-timal 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 (depend-ing 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