# Investigating and Exploiting the Bias of the Weighted Hypervolume to Articulate User Preferences

1 TAO - Machine Learning and Optimisation
LRI - Laboratoire de Recherche en Informatique, UP11 - Université Paris-Sud - Paris 11, Inria Saclay - Ile de France, CNRS - Centre National de la Recherche Scientifique : UMR8623
Abstract : Optimizing the hypervolume indicator within evolutionary multiobjective optimizers has become popular in the last years. Recently, the indicator has been generalized to the weighted case to incorporate various user preferences into hypervolume-based search algorithms. There are two main open questions in this context: (i) how does the specified weight influence the distribution of a fixed number of points that maximize the weighted hypervolume indicator? (ii) how can the user articulate her preferences easily without specifying a certain weight distribution function? In this paper, we tackle both questions. First, we theoretically investigate optimal distributions of $\mu$ points that maximize the weighted hypervolume indicator. Second, based on the obtained theoretical results, we propose a new approach to articulate user preferences within biobjective hypervolume-based optimization in terms of specifying a desired density of points on a predefined (imaginary) Pareto front. Within this approach, a new exact algorithm based on dynamic programming is proposed which selects the set of $\mu$ points that maximizes the (weighted) hypervolume indicator. Experiments on various test functions show the usefulness of this new preference articulation approach and the agreement between theory and practice.
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Conference papers
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Cited literature [19 references]

https://hal.archives-ouvertes.fr/hal-00431274
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Submitted on : Wednesday, November 11, 2009 - 3:22:07 PM
Last modification on : Sunday, June 26, 2022 - 11:50:50 AM
Long-term archiving on: : Thursday, June 17, 2010 - 8:04:57 PM

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### Citation

Anne Auger, Johannes Bader, Dimo Brockhoff, Eckart Zitzler. Investigating and Exploiting the Bias of the Weighted Hypervolume to Articulate User Preferences. Genetic and Evolutionary Computation Conference (GECCO 2009), Jul 2009, Montreal, Canada. pp.563-570, ⟨10.1145/1569901.1569980⟩. ⟨hal-00431274⟩

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