Population Diversity of Nonelitist Evolutionary Algorithms in the Exploration Phase

This paper discusses the genetic diversity of real-coded populations processed by an evolutionary algorithm (EA). Diversity is expressed as a variance or a covariance matrix of individuals contained in the population, in one- or multi-dimensional cases, respectively. We focus on the exploration stage of the optimization, therefore, the fitness function is modeled as noise. We prove that the expected value of genetic diversity achieves a level proportional to the mutation covariance matrix. The proportionality coefficient depends solely on the EA parameters. Formulas are derived to predict the diversity for fitness proportionate, tournament, and truncation selection, with and without arithmetic crossover and with Gaussian mutation. Experimental validation of the multidimensional case shows that prediction accuracy is satisfactory in a broad spectrum of settings of EA parameters.