Cone-Based Hypervolume Indicators: Construction, Properties, and Efficient Computation

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 gamma is studied. These gamma-cones can be efficiently constructed and have a number of favorable properties. It is shown that for gamma-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 mu 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 mu-distributions of this indicator. It is shown that in two dimensions and for linear Pareto fronts the optimal mu-distribution has uniform gap. For general Pareto curves and. approaching zero, it is proven that the optimal mu-distribution becomes equidistant in the Manhattan distance. An important implication of this theoretical result is that by replacing the classical hypervolume indicator by CHI with gamma-cones in hypervolume-based algorithms, such as the SMS-EMOA, the distribution can be shifted from a distribution that is focussed more on the knee point region to a distribution that is uniformly distributed. This is illustrated by numerical examples in 2-D. Moreover, in 3-D a similar dependency on gamma is observed.