The convergence of the abstract evolutionary algorithm based on a special selection mechanism

There has been a growing interest in mathematical models to character the evolutionary algorithms. The best-known one of such models is the axiomatic model called the abstract evolutionary algorithm. In this paper, we first introduce the definitions of the abstract selection and evolution operators, and that of the abstract evolutionary algorithm, which describes the evolution as an abstract stochastic process composed of these two fundamental abstract operators. In particular, a kind of abstract evolutionary algorithms based on a special selection mechanism is discussed. According to the sorting for the state space, the properties of the single step transition matrix for the algorithm are analyzed. In the end, we prove that the limit probability distribution of the Markov chains exists. The present work provides a big step toward the establishment of a unified theory of evolutionary computation.