A rigorous complexity analysis of the (1 + 1)- evolution strategy for separable functions with boolean inputs

Abstract

Evolutionary algorithms (EAs) are heuristic randomized algorithms which, by many impressive experiments, have been proven to behave quite well for optimization problems of various kinds. In order to improve our abilities in applying these algorithms three approaches should be followed in parallel. First, experiments for benchmark and practical problems have to be performed. Second, explanations about the behavior of EAs can be obtained by an analysis based on reasonable assumptions. Third, also a rigorous analysis without any unproven assumptions is necessary to establish in future a theory of EAs. Here, for the first time, a rigorous complexity analysis of the (1 + 1) evolutionary algorithm for separable functions with Boolean inputs is given. Different mutation rates are compared and the use of the crossover operator is investigated. The main contribution is not the result that the expected run time of the (1 + 1) evolutionary algorithm is O( n ln n ) for separable functions with n variables but the presentation of the methods how this result can be proven rigorously.