On the behaviour of weighted multi-recombination evolution strategies optimising noisy cigar functions

Cigar functions are convex quadratic functions that are characterised by the presence of only two distinct eigenvalues of their Hessian, the smaller one of which occurs with multiplicity one. Their ridge-like topology makes them a useful test case for optimisation strategies. This paper extends previous work on modelling the behaviour of evolution strategies with isotropically distributed mutations optimising cigar functions by considering weighted recombination as well as the effects of noise on optimisation performance. It is found that the same weights that have previously been seen to be optimal for the sphere and parabolic ridge functions are optimal for cigar functions as well. The influence of the presence of noise on optimisation performance depends qualitatively on the trajectory of the search point, which in turn is determined by the strategy's mutation strength as well as its population size and recombination weights. Analytical results are obtained for the case of cumulative step length adaptation.

[1]  Hans-Georg Beyer,et al.  On the Behaviour of Evolution Strategies Optimising Cigar Functions , 2010, Evolutionary Computation.

[2]  Haikady N. Nagaraja,et al.  18 Concomitants of order statistics , 1998, Order statistics.

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

[4]  H. Beyer,et al.  Noisy Local Optimization with Evolution Strategies , 2002 .

[5]  Stephen R. Marsland,et al.  Convergence Properties of (μ + λ) Evolutionary Algorithms , 2011, AAAI.

[6]  Nikolaus Hansen,et al.  Step-Size Adaption Based on Non-Local Use of Selection Information , 1994, PPSN.

[7]  Mohamed Jebalia.  Optimisation par stratégies d'évolution : convergence et vitesses de convergence pour des fonctions bruitées : résolution d'un problème d'identification , 2008 .

[8]  Dirk V. Arnold,et al.  Weighted Multirecombination Evolution Strategies on the Parabolic Ridge , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[9]  Arthur C. Sanderson,et al.  An approximate gaussian model of Differential Evolution with spherical fitness functions , 2007, 2007 IEEE Congress on Evolutionary Computation.

[10]  Dirk V. Arnold,et al.  On the use of evolution strategies for optimising certain positive definite quadratic forms , 2007, GECCO '07.

[11]  W. Vent,et al.  Rechenberg, Ingo, Evolutionsstrategie — Optimierung technischer Systeme nach Prinzipien der biologischen Evolution. 170 S. mit 36 Abb. Frommann‐Holzboog‐Verlag. Stuttgart 1973. Broschiert , 1975 .

[12]  Dirk V. Arnold,et al.  Step Length Adaptation on Ridge Functions , 2008, Evolutionary Computation.

[13]  Monte Lunacek.  Exploring the bias of direct search and evolutionary optimization , 2008 .

[14]  Hans-Georg Beyer,et al.  The Theory of Evolution Strategies , 2001, Natural Computing Series.

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

[16]  Hans-Georg Beyer,et al.  Evolution strategies with cumulative step length adaptation on the noisy parabolic ridge , 2008, Natural Computing.

[17]  Jens Jägersküpper,et al.  Rigorous Runtime Analysis of the (1+1) ES: 1/5-Rule and Ellipsoidal Fitness Landscapes , 2005, FOGA.

[18]  Hans-Paul Schwefel,et al.  Where Elitists Start Limping Evolution Strategies at Ridge Functions , 1998, PPSN.

[19]  L. Darrell Whitley,et al.  Ruffled by Ridges: How Evolutionary Algorithms Can Fail , 2004, GECCO.

[20]  Dirk V. Arnold,et al.  Noisy Optimization With Evolution Strategies , 2002, Genetic Algorithms and Evolutionary Computation.

[21]  Dirk V. Arnold,et al.  Optimal Weighted Recombination , 2005, FOGA.