A theoretical analysis of volume based pareto front approximations

Pradyumn Kumar Shukla, Nadja Doll, Hartmut Schmeck

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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 μ-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 inuence 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.

Original languageEnglish
Title of host publicationGECCO 2014
Subtitle of host publication Proceedings of the 2014 Genetic and Evolutionary Computation Conference
PublisherAssociation for Computing Machinery
Pages1415-1422
Number of pages8
ISBN (Print)9781450326629
DOIs
Publication statusPublished - 2014
Event16th Genetic and Evolutionary Computation Conference, GECCO 2014 - Vancouver, BC, Canada
Duration: 12 Jul 201416 Jul 2014

Publication series

NameGECCO 2014 - Proceedings of the 2014 Genetic and Evolutionary Computation Conference

Conference

Conference16th Genetic and Evolutionary Computation Conference, GECCO 2014
Country/TerritoryCanada
CityVancouver, BC
Period12/07/1416/07/14

Keywords

  • Cone domination
  • Hypervolume indicator
  • Multi-objective optimization

Fingerprint

Dive into the research topics of 'A theoretical analysis of volume based pareto front approximations'. Together they form a unique fingerprint.

Cite this