Convergence of Algorithm and the Schema Theorem in Genetic Algorithms

Abstract

In this article two aspects of GA are commented from a mathematical point of view. One is concerned with the convergence of GA, and the other is a probabilistic interpretation of the schema theorem. GA produces a stochastic process (that is, Markov chain) of populations. A week sufficient condition which guarantees the convergence to a unique stable distribution will be given which is satisfied by a wide range of current GA. On the other hand, the inequality appeared in the Schema Theorem is not possible to be interpreted from a probabilistic point of view without replacing those random variables by their expectations since the theorem includes random variables. This replacement unfortunately prevents the theorem from being true, and hence it is forced to be revised. Two types of the correct versions of the schema theorem will be given after a rigorous derivation based on a probability theory.