Abstract
The infinite population model for the genetic algorithm, where the iteration of the genetic algorithm corresponds to an iteration of a map G, is a discrete dynamical system. The map G is a composition of a selection operator and a mixing operator, where the latter models the effects of both mutation and crossover. This paper shows that for a typical mixing operator, the fixed point set of G is finite. That is, an arbitrarily small perturbation of the mixing operator will result in a map G with finitely many fixed points. Further, any sufficiently small perturbation of the mixing operator preserves the finiteness of the fixed point set.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Wright, A.H., Vose, M.D.: Stability of vertex fixed points and applications. In: Foundations of Genetic Algorithms 3, Morgan Kaufman Publishers, San Francisco (1995)
Reeves, C.R., Rowe, J.E.: Genetic Algorithms - Principles and Perspectives: A Guide to GA Theory. Kluwer Academic Publishers, Boston, MA (2003)
Vose, M.D.: The Simple Genetic Algorithm: Foundations and Theory. MIT Press, Cambridge (1999)
Wright, A.H., Bidwell, G.: A search for counterexamples to two conjecture on the simple genetic algorithm. In: Foundations of Genetic Algorithms 4, Morgan Kaufman Publishers, San Francisco (1997)
Wright, A.H., Vose, M.D.: Finiteness of the fixed point set for the simple genetic algorithm. Evolutionary Computation 3(4) (1995)
Rowe, J.E., Vose, M.D., Wright, A.H.: Group properties of crossover and mutation. Evolutionary Computation 10(2), 151–184 (2002)
Rowe, J.E.: A normed space of genetic operators with applications to scalability issues. Evolutionary Computation 9(1) (2001)
Abraham, R., Robbin, J.: Transversal Mappings and Flows. W. A. Benjamin, Inc., New York (1967)
Guillemin, V., Pollack, A.: Differential Topology. Prentice-Hall, Inc., Englewood Cliffs (1974)
Hirsch, M.: Differential Topology. Springer, Heidelberg (1976)
J.Jr., P., de Melo, W.: Geometric Theory of Dynamical Systems. Springer, Heidelberg (1982)
Jänich, K.: Vector Analysis. Springer, Heidelberg (2001)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gedeon, T., Hayes, C., Swanson, R. (2007). Genericity of the Fixed Point Set for the Infinite Population Genetic Algorithm. In: Stephens, C.R., Toussaint, M., Whitley, D., Stadler, P.F. (eds) Foundations of Genetic Algorithms. FOGA 2007. Lecture Notes in Computer Science, vol 4436. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73482-6_6
Download citation
DOI: https://doi.org/10.1007/978-3-540-73482-6_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73479-6
Online ISBN: 978-3-540-73482-6
eBook Packages: Computer ScienceComputer Science (R0)