The Multi-objective Variable Metric Evolution Strategy Part I

The covariance matrix adaptation evolution strategy (CMA-ES) is one of the most powerful evolutionary algorithms for real-valued single-objective optimization. Here a variant of the CMA-ES for multi-objective optimization (MOO) is developed. First a single-objective, elitist CMA-ES using plus-selection and step size control based on a success rule is introduced. 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 CMA-ES (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 NSDE, a multi-objective differential evolution algorithm, are experimentally shown.

[1]  Nikolaus Hansen,et al.  An Analysis of Mutative -Self-Adaptation on Linear Fitness Functions , 2006, Evolutionary Computation.

[2]  Nikolaus Hansen,et al.  The CMA Evolution Strategy: A Comparing Review , 2006, Towards a New Evolutionary Computation.

[3]  Lothar Thiele,et al.  A Tutorial on the Performance Assessment of Stochastic Multiobjective Optimizers , 2006 .

[4]  Nicola Beume,et al.  An EMO Algorithm Using the Hypervolume Measure as Selection Criterion , 2005, EMO.

[5]  R. Lyndon While,et al.  A New Analysis of the LebMeasure Algorithm for Calculating Hypervolume , 2005, EMO.

[6]  Christian Igel,et al.  Multi-objective Model Selection for Support Vector Machines , 2005, EMO.

[7]  Xiaodong Li,et al.  Solving Rotated Multi-objective Optimization Problems Using Differential Evolution , 2004, Australian Conference on Artificial Intelligence.

[8]  Nikolaus Hansen,et al.  Evaluating the CMA Evolution Strategy on Multimodal Test Functions , 2004, PPSN.

[9]  Petros Koumoutsakos,et al.  Learning probability distributions in continuous evolutionary algorithms – a comparative review , 2004, Natural Computing.

[10]  Hans-Paul Schwefel,et al.  Evolution strategies – A comprehensive introduction , 2002, Natural Computing.

[11]  Petros Koumoutsakos,et al.  Self-Adaptation for Multi-objective Evolutionary Algorithms , 2003, EMO.

[12]  Marco Laumanns,et al.  PISA: A Platform and Programming Language Independent Interface for Search Algorithms , 2003, EMO.

[13]  Petros Koumoutsakos,et al.  Reducing the Time Complexity of the Derandomized Evolution Strategy with Covariance Matrix Adaptation (CMA-ES) , 2003, Evolutionary Computation.

[14]  Gary B. Lamont,et al.  Evolutionary Algorithms for Solving Multi-Objective Problems , 2002, Genetic Algorithms and Evolutionary Computation.

[15]  Kalyanmoy Deb,et al.  A fast and elitist multiobjective genetic algorithm: NSGA-II , 2002, IEEE Trans. Evol. Comput..

[16]  KalyanmoyDebandSamirAgrawal KanpurGeneticAlgorithmsLaboratory,et al.  A Niched-Penalty Approach for Constraint Handling in Genetic Algorithms , 2002 .

[17]  Marco Laumanns,et al.  Mutation Control and Convergence in Evolutionary Multi-Object Optimization , 2001 .

[18]  Nikolaus Hansen,et al.  Completely Derandomized Self-Adaptation in Evolution Strategies , 2001, Evolutionary Computation.

[19]  D. Corne,et al.  On Metrics for Comparing Non Dominated Sets , 2001 .

[20]  Nikolaus Hansen,et al.  Invariance, Self-Adaptation and Correlated Mutations and Evolution Strategies , 2000, PPSN.

[21]  Lothar Thiele,et al.  Comparison of Multiobjective Evolutionary Algorithms: Empirical Results , 2000, Evolutionary Computation.

[22]  Kenneth V. Price,et al.  An introduction to differential evolution , 1999 .

[23]  Lothar Thiele,et al.  Multiobjective Optimization Using Evolutionary Algorithms - A Comparative Case Study , 1998, PPSN.

[24]  Peter J. Fleming,et al.  Multiobjective optimization and multiple constraint handling with evolutionary algorithms. II. Application example , 1998, IEEE Trans. Syst. Man Cybern. Part A.

[25]  J. Dennis,et al.  A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems , 1997 .

[26]  Nikolaus Hansen,et al.  On the Adaptation of Arbitrary Normal Mutation Distributions in Evolution Strategies: The Generating Set Adaptation , 1995, ICGA.

[27]  Hans-Paul Schwefel,et al.  Evolution and optimum seeking , 1995, Sixth-generation computer technology series.

[28]  M. V. Wilkes,et al.  The Art of Computer Programming, Volume 3, Sorting and Searching , 1974 .

[29]  Ingo Rechenberg,et al.  Evolutionsstrategie : Optimierung technischer Systeme nach Prinzipien der biologischen Evolution , 1973 .