Genetic algorithms: Minimal conditions for convergence

Abstract

This paper is concerning the finite, homogenous Markov chain modeling of the binary, elitist genetic algorithm (EGA) and provides a set of minimal sufficient conditions for convergence to the global optimum. The case of a GA where each population would be allow to mutate only a small number of bits has not been covered yet by the GA's literature, although it commonly appears in practice. The main result presented here shows that the condition of the one-step transition probability by mutation between two arbitrary strings being larger than zero can be relaxed in the sense that it is also sufficient to achieve the transition by a chain of small mutations. Consequently, even one-bit mutations would be sufficient to make the GA globally convergent, because they can be chained to achieve a multi-bit mutation. All this study is performed with respect to the theory of non-negative matrices and their relationship to Markov chains.

All over this paper the term elitist is associated to a canonical GA maintaining the best solution found over time, without using it to generate new individuals.