# SIMPLE GENETIC ALGORITHM WITH GENERALISED a*-SELECTION - Dynamical System Model, Fixed Points, and Schemata

### André Neubauer

#### 2009

#### Abstract

The dynamical system model proposed by VOSE provides a theory of genetic algorithms as specific random heuristic search (RHS) algorithms by describing the stochastic trajectory of a population with the help of a deterministic heuristic function and its fixed points. In order to simplify the mathematical analysis and to enable the explicit calculation of the fixed points the simple genetic algorithm (SGA) with a-selection has been introduced where the best or a-individual is mated with individuals randomly chosen from the population with uniform probability. This selection scheme also allows to derive a simple coarse-grained system model based on the equivalence relation imposed by schemata. In this paper, the a-selection scheme is generalised to a*-selection by allowing the ß best individuals of the current population instead of the single best a-individual to mate with other individuals randomly chosen from the population. It is shown that most of the results obtained for a-selection can be transferred to the SGA with generalised a*-selection, e.g. the explicit calculation of the fixed points of the heuristic function or the derivation of a coarse-grained system model based on schemata.

#### References

- Neubauer, A. (2008a). Intrinsic system model of the genetic algorithm with a-selection. In Parallel Problem Solving from Nature PPSN X, Lecture Notes in Computer Science, pages 940-949. Springer.
- Neubauer, A. (2008b). Theory of the simple genetic algorithm with a-selection. In Proceedings of the 10th Annual Genetic and Evolutionary Computation Conference - GECCO 2008, pages 1009-1016.
- Reeves, C. R. and Rowe, J. E. (2003). Genetic Algorithms - Principles and Perspectives, A Guide to GA Theory. Kluwer Academic Publishers, Boston.
- Vose, M. D. (1999a). Random heuristic search. Theoretical Computer Science, 229(1-2):103-142.
- Vose, M. D. (1999b). The Simple Genetic Algorithm - Foundations and Theory. MIT Press, Cambridge.
- Vose, M. D. and Wright, A. H. (1998). The simple genetic algorithm and the walsh transform - part i, theory - part ii, the inverse. Evolutionary Computation, 6(3):253-273, 275-289.

#### Paper Citation

#### in Harvard Style

Neubauer A. (2009). **SIMPLE GENETIC ALGORITHM WITH GENERALISED a*-SELECTION - Dynamical System Model, Fixed Points, and Schemata** . In *Proceedings of the International Joint Conference on Computational Intelligence - Volume 1: ICEC, (IJCCI 2009)* ISBN 978-989-674-014-6, pages 203-208. DOI: 10.5220/0002312602030208

#### in Bibtex Style

@conference{icec09,

author={André Neubauer},

title={SIMPLE GENETIC ALGORITHM WITH GENERALISED a*-SELECTION - Dynamical System Model, Fixed Points, and Schemata},

booktitle={Proceedings of the International Joint Conference on Computational Intelligence - Volume 1: ICEC, (IJCCI 2009)},

year={2009},

pages={203-208},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0002312602030208},

isbn={978-989-674-014-6},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Joint Conference on Computational Intelligence - Volume 1: ICEC, (IJCCI 2009)

TI - SIMPLE GENETIC ALGORITHM WITH GENERALISED a*-SELECTION - Dynamical System Model, Fixed Points, and Schemata

SN - 978-989-674-014-6

AU - Neubauer A.

PY - 2009

SP - 203

EP - 208

DO - 10.5220/0002312602030208