On the effect of selection in genetic algorithms

Abstract

To study the effect of selection with respect to mutation and mating in genetic algorithms, we consider two simplified examples in the infinite population limit. Both algorithms are modeled as measure valued dynamical systems and are designed to maximize a linear fitness on the half line. Thus, they both trivially converge to infinity. We compute the rate of their growth and we show that, in both cases, selection is able to overcome a tendency to converge to zero. The first model is a mutation-selection algorithm on the integer half line, which generates mutations along a simple random walk. We prove that the system goes to infinity at a positive speed, even in cases where the random walk itself is ergodic. This holds in several strong senses, since we show a.s. convergence, Lp convergence, convergence in distribution, and a large deviations principle for the sequence of measures. For the second model, we introduce a new class of matings, based upon Mandelbrot martingales. The mean fitness of the associated mating-selection algorithms on the real half line grows exponentially fast, even in cases where the Mandelbrot martingale itself converges to zero. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 185–200, 2001