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

  • Anne Auger
  • Johannes Bader
  • Dimo Brockhoff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6238)


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.


Extreme Point Pareto Front Objective Vector Multiobjective Evolutionary Algorithm Evolutionary Multiobjective Optimization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Anne Auger
    • 1
  • Johannes Bader
    • 2
  • Dimo Brockhoff
    • 1
  1. 1.TAO Team, INRIA Saclay, LRI, Paris Sud UniversityOrsay CedexFrance
  2. 2.Computer Engineering and Networks LabETH ZurichZurichSwitzerland

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