Markov Chain Analysis of Genetic Algorithms in a Noisy Environment

1. Abstract Genetic algorithms (GAs) have been widely proposed as an effective optimization tool for dealing with the noisy objective functions that arise in many practical problems. Although quite a few studies have examined GAs in noisy environments using either numerical or other theoretical methods, Markov chain analysis has not been conducted to investigate them (Markov chain analysis has revealed important properties of GAs in noiseless environments). We take a first step towards rigorously analyzing GAs in noisy environments using Markov chain theory. More specifically, in this study, we conduct Markov chain analysis to investigate the transition and convergence properties of GAs applied to objective functions that are additively perturbed by discrete noise. We explicitly construct a Markov chain that models these GAs and compute the transition probabilities of the chain. This chain has only one positive recurrent communication class, and it follows immediately from this property that the GAs eventually (i.e., as the number of iterations goes to infinity) find at least one globally optimal solution with probability 1. Furthermore, our analysis shows that the Markov 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 certain probability that a GA has reached the set of globally optimal solutions and continues to include in each subsequent population at least one globally optimal solution whose observed fitness value is greater than that of any suboptimal solution.