An Efficient Algorithm for Computing Hypervolume Contributions

• Published in: Evolutionary computation Vol. 18; no. 3; pp. 383 - 402
• Main Authors: Bringmann, Karl, Friedrich, Tobias
• Format: Journal Article
• Language: English
• Published: One Rogers Street, Cambridge, MA 02142-1209, USA MIT Press 01.09.2010
• Subjects: Additives; Algorithms; Biological Evolution; Computation; Criteria; Evolutionary algorithms; Indicators; Mathematical analysis; Mathematical models; Models, Statistical; Optimization; Population Density; Sorting
• ISSN: 1063-6560; 1530-9304; 1530-9304
• DOI: 10.1162/EVCO_a_00012

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 with objectives, a solution with minimal hypervolume contribution in time ( log ) for > 2. This improves all previously published algorithms by a factor of for all > 3 and by a factor of for = 3. We also analyze hypervolume indicator based optimization algorithms which remove λ > 1 solutions from a population of size = 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 with objectives, our algorithm can calculate a set of λ solutions with minimal hypervolume contribution in time ( log + ) for > 2. This improves all previously published algorithms by a factor of for > 3 and by a factor of for = 3.