The Relationship Between the Covered Fraction, Completeness and Hypervolume Indicators Viviane Grunert da Fonseca 1,3 and Carlos M. Fonseca 2,3 1 INUAF – Instituto Superior D. Afonso III, Loul´e, Portugal viviane.grunert@sapo.pt 2 CISUC, Department of Informatics Engineering University of Coimbra, Coimbra, Portugal cmfonsec@dei.uc.pt 3 CEG-IST – Center for Management Studies Instituto Superior T´ecnico, Lisbon, Portugal Abstract. This paper investigates the relationship between the covered fraction, completeness, and (weighted) hypervolume indicators for as- sessing the quality of the Pareto-front approximations produced by mul- tiobjective 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 par- ticular probability distributions. Expressions for the expected value of these distributions are derived, and shown to be directly related to the first-order attainment function. Keywords: stochastic multiobjective optimizer, performance assessment, covered fraction indicator, completeness indicator, (weighted) hypervol- ume indicator, attainment function, expected value, Hausdorff measure. 1 Introduction The performance assessment of stochastic multiobjective optimizers (MOs) has become an emerging area of research, enabling the comparison of existing op- timizers and supporting the development of new ones. In general, stochastic MO performance can be associated with the distributional behavior of the ran- dom outcomes produced in one optimization run, seen either as random non- dominated point (RNP) sets in objective space or, alternatively, as the corre- sponding random unbounded attained sets [7]. Typically, realizations of such random closed sets in R d can be observed arbitrarily often through multiple optimization runs. This allows stochastic MO performance assessment and comparison to be carried out with frequency-based statistical inference methodology using (simple) random samples of independent and identically distributed MO outcome sets. 71 2011 EA 2011 - Proceedings ISBN 978-2-9539267-1-2 Editors: Jin-Kao Hao, Pierrick Legrand, Pierre Collet, Nicolas Monmarché, Evelyne Lutton, Marc Schoenauer