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 (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 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 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 in the runtime of the calculation instead of a multiplicative factor of . 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 (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.

Note

A conference version of this article appeared under the title “Don't be greedy when calculating hypervolume contributions” (Bringmann and Friedrich, 2009a) in the Proceedings of the 10th ACM Foundations of Genetic Algorithms.

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