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Tight bounds for the approximation ratio of the hypervolume indicator (2010)
| Venue: | In Proc. 11th International Conference 29 Problem Solving from Nature (PPSN XI), volume 6238 of LNCS |
| Citations: | 7 - 5 self |
BibTeX
@INPROCEEDINGS{Bringmann10tightbounds,
author = {Karl Bringmann and Tobias Friedrich},
title = {Tight bounds for the approximation ratio of the hypervolume indicator},
booktitle = {In Proc. 11th International Conference 29 Problem Solving from Nature (PPSN XI), volume 6238 of LNCS},
year = {2010},
pages = {607--616},
publisher = {Springer}
}
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Abstract
Abstract The hypervolume indicator is widely used to guide the search and to evaluate the performance of evolutionary multi-objective optimization algorithms. It measures the volume of the dominated portion of the objective space which is considered to give a good approximation of the Pareto front. There is surprisingly little theoretically known about the quality of this approximation. We examine the multiplicative approximation ratio achieved by two-dimensional sets maximizing the hypervolume indicator and prove that it deviates significantly from the optimal approximation ratio. This provable gap is even exponential in the ratio between the largest and the smallest value of the front. We also examine the additive approximation ratio of the hypervolume indicator and prove that it achieves the optimal additive approximation ratio apart from a small factor � n/(n − 2), where n is the size of the population. Hence the hypervolume indicator can be used to achieve a very good additive but not a good multiplicative approximation of a Pareto front. 1
Keyphrases
hypervolume indicator tight bound approximation ratio pareto front evolutionary multi-objective optimization algorithm optimal additive approximation good multiplicative approximation good approximation dominated portion objective space provable gap optimal approximation ratio multiplicative approximation ratio additive approximation ratio good additive two-dimensional set small factor
