Theory of genetic algorithms

Abstract

(i) We investigate spectral and geometric properties of the mutation-crossover operator in a genetic algorithm with general-size alphabet. By computing spectral estimates, we show how the crossover operator enhances the averaging procedure of the mutation operator in the random generator phase of the genetic algorithm. By mapping our model to the multi-set model often investigated in the literature, we compute corresponding spectral estimates for mutation-crossover in the multi-set model.(ii) Various types of unscaled or scaled fitness selection mechanisms are considered such as proportional fitness selection, rank selection, and tournament fitness selection. We allow fitness selection mechanisms where the fitness of an individual or creature depends upon the population it resides in. We investigate contracting properties of these fitness selection mechanisms and combine them with the crossover operator to obtain a model for genetic drift. This has applications to the study of genetic algorithms with zero or extremely low mutation rate.(iii) We discuss a variety of convergent simulated-annealing-type algorithms with mutation-crossover as generator matrix.(iv) The theory includes proof of strong ergodicity for various types of scaled genetic algorithms using common fitness selection methods. If the mutation rate converges to a positive value, and the other operators of the genetic algorithm converge, then the limit probability distribution over populations is fully positive at uniform populations whose members have not necessarily optimal fitness.(v) In what follows, suppose the mutation rate converges to zero sufficiently slow to assure weak ergodicity of the inhomogeneous Markov chain describing the genetic algorithm, unbounded power-law scaling for the fitness selection is used, mutation and crossover commute, and the fitness function is injective which is a minor restriction in regard to function optimization.(va) If a certain integrable convergence condition is satisfied such that the selection pressure increases fast, then there is essentially no other restriction on the crossover operation, and the algorithm asymptotically behaves as the following take-the-best search algorithm: (1) mutate in every step with rate decreasing to zero, and (2) map any population to the uniform population with the best creature. The take-the-best search algorithm is investigated, and its convergence is shown. Depending upon population-size, the take-the-best search algorithm does or does not necessarily converge to the optimal solution.(vb) If population size is larger than length of genome, and a certain logarithmic convergence condition is satisfied such that the selection pressure increases slowly but sufficiently fast, then the algorithm asymptotically converges to the optimal solution.