Evolutionary Multiobjective Optimization Algorithm as a Markov System

In the paper we consider the ranking given by the Pareto dominance relation as a basis to create a selection operator for the Evolutionary Multiobjective Optimization Algorithm (EMOA). Assuming that sampling to the next epoch is performed according to the generalized Bernoulli schema with regard to a selected type of the rank selection, a heuristic operator for EMOA is introduced. Having defined the heuristic operator, the transition probability matrix of the uniform Markov chain modeling EMOA can be explicitly obtained as in the Vose's theory of the Simple Genetic Algorithm (SGA). This chain is ergodic if the mixing operator following the EMOA selection operator in each epoch is strictly positive. Moreover, we show that the measure on the space of populations imposed by the EMOA infinite population concentrates on the set of fixed points of the heuristic operator after infinite number of epochs, assuming that the heuristic operator is focusing.