Fractals and Evolutionary Algorithms
First interactions between Evolutionary Algorithms and Fractals were for the resolution of some difficult optimisation problems encountered in Fractal Analysis: Among them the famous inverse problem for Iterated Function Systems, which is a challenging problem in image and signal compression [3,4,7,17,21,24]. However Fractal analysis provides a theoretical tool that help to understand some aspects of EA behaviour [13,14,16,18,23].
As usual in such domains there are two main approaches to understand the complex dynamic behaviour of these algorithms:
To make as few as possible hypotheses on the functions to be optimised in order to obtain some global (simple) results. Markov models, [2,5,10,20], Schema theory [8,9,11,12] belong to this item.
To obtain some more precise results by setting restrictive hypotheses on the fitness function : NK Landscapes [19], Evolutionary Strategies on sphere models [1], for example.
If we investigate how fractal irregularity analysis can be used to refine some theoretical analysis of EA, in the framework of this second item, i.e. when some assumptions on the irregularity of the fitness function are made:
Some results [16] relate the irregularity of the fitness function to its ``difficulty'' (according to the usual intuition of EA practitioner), and provide information about how to modify some of the GA parameters in order to improve its performances. An important application of such a theoretical analysis provides a mean of measuring (of course to a certain extent, due to the intrinsic limitations of the model) the influence of the chromosome encoding. Some experimentations with the Gray encoding have also been performed [15].
Then in [14] a preliminary analysis has been developed, derived from Freidlin-Wentzell theory, related to the convergence of a simple EA model. This opens the way to convergence speed bounds with respect to some statistical measures on the fitness function (likely related to irregularity).
Fractal irregularity analysis has been proven useful in various domains (physics, finance, image and signal analysis, see [6,22]). It is now clear that it has open the way to new theoretical investigations for the analysis of EA behaviour. This allow us to suggest that irregularity analysis may be a tool of interest for the theoretical investigation of other stochastic optimisation heuristics.
See also the C code for the fractal test functions
References
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See also (in the publications list) :
Perturbation theory for EAs: towards an estimation of convergence speed |
Yann Landrin-Schweitzer and Evelyne Lutton |
In PPSN VI, Paris - France, Sept 16-20, 2000. |
Genetic Algorithms and Fractals |
Evelyne Lutton |
In Evolutionary Algorithms in Engineering and Computer Science, 1999. |
Genetic Algorithms and Fractals - Algorithmes Génétiques et Fractales |
Evelyne Lutton |
Dossier d'Habilitation à diriger des recherches, Université Paris XI Orsay, Spécialité Informatique, 11 Février, 1999. |
Holder functions and Deception of Genetic Algorithms |
Jacques Levy Vehel and Evelyne Lutton |
In IEEE transactions on Evolutionary computing, No. 2, Vol. 2, July, 1998. |
Bitwise regularity and GA-hardness |
Benoit Leblanc and Evelyne Lutton |
In CEC 98, Anchorage, Alaska, May 5-9, 1998. |
Bitwise Regularity Coefficients as a Tool for Deception Analysis of a Genetic Algorithm |
Benoit Leblanc and Evelyne Lutton |
INRIA Rocquencourt, RR-3274, 1997. |
Some remarks on the optimization of Hölder functions with Genetic Algorithms |
Jacques Levy Vehel and Evelyne Lutton |
INRIA Rocquencourt, RR-2627, 35 p., July, 1995. |
Optimization of Fractal Functions Using Genetic Algorithms |
Jacques Levy Vehel and Evelyne Lutton |
In Fractal'93, London, 1993. |