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Hyperbolic fixed points are typical in the space of mixing operators for the infinite population genetic algorithm

Published: 25 June 2005 Publication History

Abstract

We study an infinite population model for the genetic algorithm, where the iteration of the algorithm corresponds to an iteration of a map G. The map G is a composition of a selection operator and a mixing operator, where the latter models effects of both mutation and crossover. We examine the hyperbolicity of fixed points of this model. We show that for a typical mixing operator all the fixed points are hyperbolic.

References

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C. Hayes and T. Gedeon. Hyperbolic fixed points are typical in the space of mixing operators for the infinite population genetic algorithm. Technical Report, http://www.math.montana.edu/~gedeon/gen_alg.html.
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S. Lang. Complex Analysis. Springer, 4th edition, 1999.
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C. R. Reeves and J. E. Rowe. Genetic Algorithms - Principles and Perspectives: A Guide to GA Theory. Kluwer Academic Publishers, 2003.
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C. Robinson. Dynamical Systems: Stability, Symbolic Dynamics and Chaos. CRC Press, 1995.
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J. E. Rowe, M. D. Vose, and A. H. Wright. Group properties of crossover and mutation. Evolutionary Computation, 10(2):151--184, 2002.
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M. D. Vose. The Simple Genetic Algorithm: Foundations and Theory. The MIT Press, 1999.
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M. D. Vose and A. H. Wright. Simple genetic algorithms with linear fitness. Evolutionary Computation, 2(4):347--368, 1994.
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A. H. Wright and G. Bidwell. A search for counterexamples to two conjectures on the simple genetic algorithm. Foundations of Genetic Algorithms 4, Morgan Kaufman Publishers, 1997.
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A. H. Wright and J. E. Rowe. Continuous dynamical systems models of steady-state genetic algorithms. Foundations of Genetic Algorithms, 6, Morgan Kaufmann, 2001.
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A. H. Wright and M. D. Vose. Stability of vertex fixed points and applications. Foundations of Genetic Algorithms 3, Morgan Kaufman Publishers, 1995.

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  1. Hyperbolic fixed points are typical in the space of mixing operators for the infinite population genetic algorithm

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        cover image ACM Conferences
        GECCO '05: Proceedings of the 7th annual workshop on Genetic and evolutionary computation
        June 2005
        431 pages
        ISBN:9781450378000
        DOI:10.1145/1102256
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        Published: 25 June 2005

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        Author Tags

        1. artificial intelligence
        2. generic
        3. genetic algorithm
        4. hyperbolic fixed point
        5. mixing
        6. typical

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