An Efficient Algorithm for Computing Hypervolume Contributions

Abstract

The hypervolume indicator serves as a sorting criterion in many recent multi-objective evolutionary algorithms (MOEAs). Typical algorithms remove the solution with the smallest loss with respect to the dominated hypervolume from the population. We present a new algorithm which determines for a population of size n with d objectives, a solution with minimal hypervolume contribution in time [Formula: see text](nd/2 log n) for d > 2. This improves all previously published algorithms by a factor of n for all d > 3 and by a factor of [Formula: see text] for d = 3. We also analyze hypervolume indicator based optimization algorithms which remove λ > 1 solutions from a population of size n = μ + λ. We show that there are populations such that the hypervolume contribution of iteratively chosen λ solutions is much larger than the hypervolume contribution of an optimal set of λ solutions. Selecting the optimal set of λ solutions implies calculating [Formula: see text] conventional hypervolume contributions, which is considered to be computationally too expensive. We present the first hypervolume algorithm which directly calculates the contribution of every set of λ solutions. This gives an additive term of [Formula: see text] in the runtime of the calculation instead of a multiplicative factor of [Formula: see text]. More precisely, for a population of size n with d objectives, our algorithm can calculate a set of λ solutions with minimal hypervolume contribution in time [Formula: see text](nd/2 log n + nλ) for d > 2. This improves all previously published algorithms by a factor of nmin{λ,d/2} for d > 3 and by a factor of n for d = 3.