On Stability and Classification Tools for Genetic Algorithms

Convergence of genetic algorithms in the form of asymptotic stability requirements is discussed. Some tools to measure convergence properties of genetic algorithms are introduced. A classification procedure is proposed that is based on the following conjecture: the entropy and the fractal dimension of trajectories of genetic algorithms produced by them are quantities that can characterize the algorithms. The role of these quantities as invariants of the algorithm classes is discussed together with the compression ratio of points of genetic algorithms.

[1]  W. Szlenk,et al.  An introduction to the theory of smooth dynamical systems , 1984 .

[2]  D. Ornstein,et al.  On isomorphism of weak Bernoulli transformations , 1970 .

[3]  Witold Kosinski,et al.  On Convergence of a Simple Genetic Algorithm , 2008, ICAISC.

[4]  Thomas M. English.  No more lunch: analysis of sequential search , 2004, Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No.04TH8753).

[5]  Y. Suhov,et al.  An entropic measurement of queueing behaviour in a class of manufacturing operations , 2001, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[6]  D. Ornstein.  Ergodic theory, randomness, and dynamical systems , 1974 .

[7]  J. E. Rowe.  The dynamical systems model of the simple genetic algorithm , 2001 .

[8]  Robert Schaefer,et al.  Foundations of Global Genetic Optimization , 2007, Studies in Computational Intelligence.

[9]  Marc Toussaint,et al.  A No-Free-Lunch theorem for non-uniform distributions of target functions , 2004, J. Math. Model. Algorithms.

[10]  Zbigniew Michalewicz,et al.  Genetic Algorithms + Data Structures = Evolution Programs , 1996, Springer Berlin Heidelberg.

[11]  David H. Wolpert,et al.  No free lunch theorems for optimization , 1997, IEEE Trans. Evol. Comput..

[12]  Zbigniew Michalewicz,et al.  Fractal Dimension of Trajectory as Invariant of Genetic Algorithms , 2006, ICAISC.

[13]  Adam Hammett.  An Introduction to Fractals , 2011 .

[14]  Michael D. Vose,et al.  Modeling Simple Genetic Algorithms , 1995, Evolutionary Computation.

[15]  Geon Ho Choe,et al.  Computational Ergodic Theory , 2005 .