The relationship between the covered fraction, completeness and hypervolume indicators

Authors:
Viviane Grunert da Fonseca;Carlos M. Fonseca
Affiliations:
INUAF --- Instituto Superior D. Afonso III, Loulé, Portugal, CEG-IST --- Center for Management Studies, Instituto Superior Técnico, Lisbon, Portugal;CISUC, Department of Informatics Engineering, University of Coimbra, Coimbra, Portugal, CEG-IST --- Center for Management Studies, Instituto Superior Técnico, Lisbon, Portugal
Venue:
EA'11 Proceedings of the 10th international conference on Artificial Evolution
Year:
2011

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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.