An asymptotic theory for genetic algorithms

Abstract

The Freidlin-Wentzell theory deals with the study of random perturbations of dynamical systems. We build several models of genetic algorithms by randomly perturbing simple processes. The asymptotic dynamics of the resulting processes is analyzed with the powerful tools developed by Freidlin and Wentzell and later by Azencott, Catoni and Trouvé in the framework of the generalized simulated annealing. First, a markovian model inspired by Holland's simple genetic algorithm is built by randomly perturbing a very simple selection scheme. The convergence toward the global maxima of the fitness function becomes possible when the population size is greater than a critical value which depends strongly on the optimization problem. In the bitstring case, the critical value of the population size is smaller than a linear function of the chromosome length, provided that the difficulty of the fitness landscape remains bounded. We then use the concepts introduced by Catoni and further generalized by Trouvé to fathom more deeply the dynamics of the two operators mutation-selection algorithm when the population size becomes very large. A new genetic algorithm is finally presented. A new selection mechanism is used, which has the decisive advantage of preserving the diversity of the individuals in the population. When the population size is greater than a critical value, the delicate asymptotic interaction of the perturbations ensures the convergence toward a set of populations which contain all the global maxima of the fitness function.