Application of the lower-bound function method to the investigation of the convergence of genetic algorithms

Markovian operators, non-negative linear operators and its subgroups play a significant role for the description of phenomena observed in the nature. Research on asymptotic stability is one of the main issues in this respect. A. Lasota and J. A. Yorke proved in 1982 that the necessary and sufficient condition of the asymptotic stability of a Markovian operator is the existence of a non-trivial lower-bound function. In the present paper it is shown how the method of lower-bound function can be applied to the investigation of genetic algorithms. Genetic algorithms considered used for solving of non-smooth optimization problems are compositions of two random operators: selection and mutation. The compositions are Markovian matrices.