Abstract
The simple genetic algorithm (SGA) and its convergence analysis are main subjects of the article. The SGA is defined on a finite multi-set of potential problem solutions (individuals) together with mutation and selection operators, and appearing with some prescribed probabilities. The selection operation acts on the basis of the fitness function defined on individuals, and is fundamental for the problem considered. Generation of new population is realized by iterative actions of those operators written in the form of a transition operator acting on probability vectors. The transition operator is a Markov one. Conditions for convergence and asymptotic stability of the transition operator are formulated.
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
Kieśi, P., Michalewicz, Z.: Foundations of Genetic Algorithms (in Polish). Matematyka Stosowana. Matematyka dla Społeczeństwa, Applied Mathematics. Mathematics for Society 1(44), 68–91 (2000)
Kotowski, S., Socała, J., Kosiński, W., Michalewicz, Z.: Markovian model of simple genetic algorithms and its asymptotic behaviour (under preparation)
Lasota, A.: Asymptotic properties of Markov operators semigroups(in Polish). Matematyka Stosowana. Matematyka dla Społeczeństwa, Applied Mathematics. Mathematics for Society 3(46), 39–51 (2002)
Lasota, A., Yorke, J.A.: Exact dynamical systems and the Frobenius–Perron operator. Trans. Amer. Math. Soc. 273, 375–384 (1982)
Michalewicz, Z.: Genetic Algorithms + Data Structures = Evolution Programs, 3rd edn. Springer, Heidelberg (1996)
Rowe, J.E.: The dynamical system models of the simple genetic algorithm. In: Kallel, L., Naudts, B., Rogers, A. (eds.) Theoretical Aspects of Evolutionary Computing, pp. 31–57. Springer, Heidelberg (2001)
Rudnicki, R.: On asymptotic stability and sweeping for Markov operators. Bull. Polish Acad. Sci. Math. 43, 245–262 (1995)
Schaefer, R.: Foundations of genetic global optimization (in Polish). In: Podstawy genetycznej optymalizacji globalnej, Jagiellonian University Press, Cracow (2002)
Socała, J.: Asymptotic behaviour of the iterates of nonnegative operators on a Banach lattice. Ann. Polon. Math. 68(1), 1–16 (1998)
Socała, J., Kosiński, W., Kotowski, S.: On asymptotic behaviour of a simple genetic algorithms (in Polish). Matematyka Stosowana. Matematyka dla Społeczeństwa, Applied Mathematics. Mathematics for Society 47, 70–86 (2005)
Socała, J., Kosiński, W.: Zastosowanie metody funkcji dolnej do badania zbieżności algorytmów genetycznych (in Polish). Matematyka Stosowana. Matematyka dla Społeczeństwa, Applied Mathematics. Mathematics for Society 8(49), 23–36 (2007)
Vose, M.D.: The Simple Genetic Algorithm: Foundation and Theory. MIT Press, Cambridge (1999)
Vose, M.D.: Modelling Simple Genetic Algorithms. Evolutionary Computation 3(4), 453–472 (1996)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Socała, J., Kosiński, W. (2008). On Convergence of a Simple Genetic Algorithm. In: Rutkowski, L., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds) Artificial Intelligence and Soft Computing – ICAISC 2008. ICAISC 2008. Lecture Notes in Computer Science(), vol 5097. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69731-2_48
Download citation
DOI: https://doi.org/10.1007/978-3-540-69731-2_48
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69572-1
Online ISBN: 978-3-540-69731-2
eBook Packages: Computer ScienceComputer Science (R0)