A Markov chain that models genetic algorithms in noisy environments

Abstract

In this study, we construct a Markov chain that models genetic algorithms (GAs) in noisy environments. Fitness functions are assumed to be perturbed by additive or multiplicative random noise that takes on finitely many values. We first compute the transition probabilities of this Markov chain exactly. We then focus on the environment where fitness functions are additively disturbed and further analyze the chain in order to investigate the transition and convergence properties of GAs in this noisy environment. The Markov chain has only one positive recurrent communication class, and it follows immediately from this property that GAs eventually find at least one globally optimal solution with probability 1. Furthermore, our analysis shows that the chain has a stationary distribution that is also its steady-state distribution. Using this property and the transition probabilities of the chain, we derive an upper bound for the number of iterations sufficient to ensure with at least a specified probability that a GA selects a globally optimal solution upon termination.