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Theoretically Investigating Optimal μ-Distributions for the Hypervolume Indicator: First Results for Three Objectives

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Parallel Problem Solving from Nature, PPSN XI (PPSN 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6238))

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Abstract

Several indicator-based evolutionary multiobjective optimization algorithms have been proposed in the literature. The notion of optimal μ-distributions formalizes the optimization goal of such algorithms: find a set of μ solutions that maximizes the underlying indicator among all sets with μ solutions. In particular for the often used hypervolume indicator, optimal μ-distributions have been theoretically analyzed recently. All those results, however, cope with bi-objective problems only. It is the main goal of this paper to extend some of the results to the 3-objective case. This generalization is shown to be not straight-forward as a solution’s hypervolume contribution has not a simple geometric shape anymore in opposition to the bi-objective case where it is always rectangular. In addition, we investigate the influence of the reference point on optimal μ-distributions and prove that also in the 3-objective case situations exist for which the Pareto front’s extreme points cannot be guaranteed in optimal μ-distributions.

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Auger, A., Bader, J., Brockhoff, D. (2010). Theoretically Investigating Optimal μ-Distributions for the Hypervolume Indicator: First Results for Three Objectives. In: Schaefer, R., Cotta, C., Kołodziej, J., Rudolph, G. (eds) Parallel Problem Solving from Nature, PPSN XI. PPSN 2010. Lecture Notes in Computer Science, vol 6238. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15844-5_59

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  • DOI: https://doi.org/10.1007/978-3-642-15844-5_59

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15843-8

  • Online ISBN: 978-3-642-15844-5

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