Finiteness of the fixed point set for the simple genetic algorithm

Authors:
Alden H. Wright;Michael D. Vose
Affiliations:
Department of Computer Science The University of Montana Missoula, MT 59812-1008 wright@cs.umt.edu;Department of Computer Science The University of Tennessee Knoxville, TN 37996-1301 vose@cs.utk.edu
Venue:
Evolutionary Computation
Year:
1995

Citing 2
Cited 2

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Simple genetic algorithms with linear fitness

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Fundamenta Informaticae

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Abstract

The infinite population simple genetic algorithm is a discrete dynamical system model of a genetic algorithm. It is conjectured that trajectories in the model always converge to fixed points. This paper shows that an arbitrarily small perturbation of the fitness will result in a model with a finite number of fixed points. Moreover, every sufficiently small perturbation of fimess preserves the finiteness of the fixed point set. These results allow proofs and constructions that require finiteness of the fixed point set. For example, applying the stable manifold theorem to a fixed point requires the hyperbolicity of the differential of the transition map of the genetic algorithm, which requires (among other things) that the fixed point be isolated.