Covariance Matrix Adaptation for Multi-objective Optimization

  • Authors:

  • Christian Igel;Nikolaus Hansen;Stefan Roth

  • Affiliations:

  • Institut für Neuroinformatik, Ruhr-Universität Bochum, 44780 Bochum, Germany christian.igel@neuroinformatik.rub.de;Computational Laboratory (CoLab), Institute of Computational Science, ETH Zurich, 8092 Zurich, Switzerland nikolaus.hansen@inf.ethz.ch;Institut für Neuroinformatik, Ruhr-Universitüt Bochum, 44780 Bochum, Germany stefan.roth@neuroinformatik.rub.de

  • Venue:
  • Evolutionary Computation
  • Year:
  • 2007

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Abstract

The covariancematrix adaptation evolution strategy (CMA-ES) is one of themost powerful evolutionary algorithms for real-valued single-objective optimization. In this paper, we develop a variant of the CMA-ES for multi-objective optimization (MOO). We first introduce a single-objective, elitist CMA-ES using plus-selection and step size control based on a success rule. This algorithm is compared to the standard CMA-ES. The elitist CMA-ES turns out to be slightly faster on unimodal functions, but is more prone to getting stuck in sub-optimal local minima. In the new multi-objective CMAES (MO-CMA-ES) a population of individuals that adapt their search strategy as in the elitist CMA-ES is maintained. These are subject to multi-objective selection. The selection is based on non-dominated sorting using either the crowding-distance or the contributing hypervolume as second sorting criterion. Both the elitist single-objective CMA-ES and the MO-CMA-ES inherit important invariance properties, in particular invariance against rotation of the search space, from the original CMA-ES. The benefits of the new MO-CMA-ES in comparison to the well-known NSGA-II and to NSDE, a multi-objective differential evolution algorithm, are experimentally shown.