Covariance Matrix Adaptation for Multi-objective Optimization

• Published in: Evolutionary computation Vol. 15; no. 1; pp. 1 - 28
• Main Authors: Igel, Christian, Hansen, Nikolaus, Roth, Stefan
• Format: Journal Article
• Language: English
• Published: One Rogers Street, Cambridge, MA 02142-1209, USA MIT Press 01.03.2007
• Subjects: Algorithms; Analysis of Variance; Biological Evolution; Computer Simulation; covariance matrix adaptation; evolution strategy; Models, Genetic; Multi-objective optimization
• ISSN: 1063-6560; 1530-9304
• DOI: 10.1162/evco.2007.15.1.1

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.