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In this paper we discuss cone-based hypervolume indicators (CHI) that generalize the classical hypervolume indicator (HI) in Pareto optimization. A family of polyhedral cones with scalable opening angle γ is studied. These γ-cones can be efficiently constructed and have a number of favorable properties. It is shown that for γ-cones dominance can be checked efficiently and the CHI computation can be reduced to the computation of the HI in linear time with respect to the number of points μ in an approximation set. Besides, individual contributions to these can be computed using a similar transformation to the case of Pareto dominance cones.

Furthermore, we present first results on theoretical properties of optimal μ-distributions of this indicator. It is shown that in two dimensions and for linear Pareto fronts the optimal μ-distribution has uniform gap. For general Pareto curves and γ approaching zero, it …