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The Relationship between the Covered Fraction, Completeness and Hypervolume Indicators

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Artificial Evolution (EA 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7401))

Abstract

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.

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Author information

Authors and Affiliations

  1. INUAF – Instituto Superior D. Afonso III, Loulé, Portugal

    Viviane Grunert da Fonseca

  2. CISUC, Department of Informatics Engineering, University of Coimbra, Coimbra, Portugal

    Carlos M. Fonseca

  3. CEG-IST – Center for Management Studies, Instituto Superior Técnico, Lisbon, Portugal

    Viviane Grunert da Fonseca & Carlos M. Fonseca

Authors
  1. Viviane Grunert da Fonseca
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  2. Carlos M. Fonseca
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Editor information

Editors and Affiliations

  1. LERIA, Laboratoire d’Etude et de echerche en Informatique d’Angers, Université d’Angers, 2 Boulevard Lavoisier, 49045, Angers Cedex 01, France

    Jin-Kao Hao

  2. Equipe ALEA, INRIA Bordeaux sud-ouest, Institut de Mathématiques de Bordeaux (IMB), UMR CNRS 5251, Université Bordeaux Segalen, 3ter place de la victoire, 33076, Bordeaux, France

    Pierrick Legrand

  3. Laboratoire des Sciences de l’Ingénieur, de l’Informatique et de l’Imagerie (ICUBE), Part d’innovation, University of Strasbourg, Boulevard Sébastien Brant, BP 10413, 67412, Illkirch Cedex, France

    Pierre Collet

  4. Ecole Polytechnique de l’Université de Tours, Laboratoire d’Informatique de l’Université de Tours, 64 avenue Jean Portalis, 37200, Tours, France

    Nicolas Monmarché

  5. INRIA Saclay - Île-de-France, Equipe AVIZ, Bât 490, 91405, Orsay Cedex, France

    Evelyne Lutton

  6. INRIA Saclay - Ile de France, Equipe TAO, LRI, Université Paris Sud, Bât 490, 91405, Orsay Cedex, France

    Marc Schoenauer

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Grunert da Fonseca, V., Fonseca, C.M. (2012). The Relationship between the Covered Fraction, Completeness and Hypervolume Indicators. In: Hao, JK., Legrand, P., Collet, P., Monmarché, N., Lutton, E., Schoenauer, M. (eds) Artificial Evolution. EA 2011. Lecture Notes in Computer Science, vol 7401. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35533-2_3

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  • DOI: https://doi.org/10.1007/978-3-642-35533-2_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-35532-5

  • Online ISBN: 978-3-642-35533-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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