On a Population Sizing Model for Evolution Strategies in Multimodal Landscapes

This article derives a population sizing model for standard evolution strategies (ES) in highly multimodal fitness landscapes with exponentially many local optima. The Rastrigin, Bohachevsky, and Ackley test functions are considered. Due to the highly nonconvex structure of these functions a detailed analytical description of the behavior of the ES is a challenge. Therefore, a model is derived that simplifies the complex structure of the functions under consideration. The main idea of this model is the interpretation of local landscape oscillations as frozen noise. This allows for an estimation of the success probability of the ES converging to the global optimum and in turn an estimation of the population size required. It is shown that the population size scales usually sublinearly with the search space dimension N. For the Rastrigin and Bohachevsky function, the population size scales with ${\mathcal {O}}(\sqrt {N} \ln (N))$ . As for Ackley, the scaling behavior depends strongly on the initial values. If the algorithm starts in a certain vicinity of the global optimizer, the dependence on the dimension N is rather weak. However, if the initial value exceeds a certain distance R to the optimizer, the population size scales exponentially with R.