Novel Algorithm to Calculate Hypervolume Indicator of Pareto Approximation Set

Hypervolume indicator is a commonly accepted quality measure for comparing Pareto approximation set generated by multi-objective optimizers. The best known algorithm to calculate it for n points in ddimensional space has a run time of O(n) with special data structures. This paper presents a recursive, vertex-splitting algorithm for calculating the hypervolume indicator of a set of n non-comparable points in d > 2 dimensions. It splits out multiple child hyper-cuboids which can not be dominated by a splitting reference point. In special, the splitting reference point is carefully chosen to minimize the number of points in the child hyper-cuboids. The complexity analysis shows that the proposed algorithm achieves O(( d 2 )) time and O(dn) space complexity in the worst case.