NDLI: A theoretical analysis of volume based Pareto front approximations

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A theoretical analysis of volume based Pareto front approximations

Content Provider	ACM Digital Library
Author	Schmeck, Hartmut Doll, Nadja Shukla, Pradyumn Kumar
Abstract	Many multi-objective algorithms use volume based quality indicators to approximate the Pareto front. Amongst these, the hypervolume is the most widely used. The distribution of solution sets of finite size μ that maximize the hypervolume have been investigated theoretically. But nearly all results are limited to the bi-objective case. In this paper, many of these results are extended to higher dimensions and a theoretical analysis and characterization of optimal \$\mu\$-distributions is done. We investigate monotonic Pareto curves that are embedded in three and higher dimensions that keep the property of the bi-objective case that only few points are determining the hypervolume contribution of a point. For finite μ, we consider the influence of the choice of the reference point and determine sufficient conditions that assure the extreme points of the Pareto curves to be included in an optimal μ- distribution. We state conditions about the slope of the front that makes it impossible to include the extremes. Furthermore, we prove more specific results for three dimensional linear Pareto fronts. It is shown that the equispaced property of an optimal distribution for a line in two dimensions does not hold in higher dimensions. We additionally investigate hypervolume in general dimensions and problems with cone domination structures.
Starting Page	1415
Ending Page	1422
Page Count	8
File Format	PDF
ISBN	9781450326629
DOI	10.1145/2576768.2598348
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2014-07-12
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Cone domination Hypervolume indicator Multi-objective optimization
Content Type	Text
Resource Type	Article

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