Theoretical analysis of genetic algorithms in noisy environments based on a Markov Model

Abstract

In this study, we take a first step towards theoretically analyzing genetic algorithms (GAs) in noisy environments using Markov chain theory. We explicitly construct a Markov chain that models GAs applied to fitness functions perturbed by either additive or multiplicative noise that takes on finitely many values, and we analyze the chain to investigate the transition and convergence properties of the GAs. For the additive case, our analysis shows that GAs eventually (i.e., as the number of iterations goes to infinity) find at least one globally optimal solution with probability 1. In contrast, GAs may eventually with probability 1 fail to do so in the multiplicative case, and we establish a condition that is both necessary and sufficient for eventually finding a globally optimal solution. In addition, our analysis shows that the chain has a stationary distribution that is also its steady-state distribution. Based on this property, 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.

Markov chain analysis of genetic algorithms applied to fitness functions perturbed concurrently by additive and multiplicative noise

“…Note that if the multiplicative noise is 1 with probability 1, then fitness functions are perturbed only by the additive noise [see (3.1)]. Therefore we obtain the following corollary, which was first reported by Nakama [15]:…”

“…Nakama [14,15] constructed a Markov chain that explicitly models GAs in noisy environments, and we will closely follow and extend his notation in this paper. Our notation becomes complicated due to the inclusion of two noise sources; we will use simple examples to explain it.…”

“…However, this Markov chain fails to capture the evolution of GAs in noisy environments. Instead, we follow the Markov chain construction by Nakama [14,15] and establish a chain, call it (Z k ), whose state space consists of multisets not of chromosomes but of the ordered triples defined below.…”

“…Multiplicative noise has been considered in a variety of practical optimization problems, such as control optimization (e.g., Hopkins [7]), image restoration (e.g., Rudin and Osher [19]), and portfolio optimization (e.g., Primbs [18]). Nakama [15] was the first to theoretically investigate the properties of GAs with multiplicative noise (see also Jabelia and Auger [8]). For the multiplicative noise case, the degree of fitness disturbance may not be the same at all points in the search space.…”

Log-Linear Convergence and Divergence of the Scale-Invariant (1+1)-ES in Noisy Environments

“…Note that other rigorous theoretical studies exist, however restricted to a discrete and finite search space [25,26].…”