The Relationship between the Covered Fraction, Completeness and Hypervolume Indicators

  • Viviane Grunert da Fonseca
  • Carlos M. Fonseca
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7401)


This paper investigates the relationship between the covered fraction, completeness, and (weighted) hypervolume indicators for assessing the quality of the Pareto-front approximations produced by multiobjective optimizers. It is shown that these unary quality indicators are all, by definition, weighted Hausdorff measures of the intersection of the region attained by such an optimizer outcome in objective space with some reference set. Moreover, when the optimizer is stochastic, the indicators considered lead to real-valued random variables following particular probability distributions. Expressions for the expected value of these distributions are derived, and shown to be directly related to the first-order attainment function.


stochastic multiobjective optimizer performance assessment covered fraction indicator completeness indicator (weighted) hypervolume indicator attainment function expected value Hausdorff measure 


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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Viviane Grunert da Fonseca
    • 1
    • 3
  • Carlos M. Fonseca
    • 2
    • 3
  1. 1.INUAF – Instituto Superior D. Afonso IIILouléPortugal
  2. 2.CISUC, Department of Informatics EngineeringUniversity of CoimbraCoimbraPortugal
  3. 3.CEG-IST – Center for Management StudiesInstituto Superior TécnicoLisbonPortugal

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