On a Multi–Objective Evolutionary Algorithm and Its Convergence to the Pareto Set

Although there are many versions of evolutionary algorithms that are tailored to multi–criteria optimization, theoretical results are apparently not yet available. Here, it is shown that results known from the theory of evolutionary algorithms in case of single criterion optimization do not carry over to the multi–criterion case. At first, three different step size rules are investigated numerically for a selected problem with two conflicting objectives. The empirical results obtained by these experiments lead to the observation that only one of these step size rules may have the property to ensure convergence to the Pareto set. A theoretical analysis finally shows that a special version of an evolutionary algorithm with this step size rule converges with probability one to the Pareto set for the test problem under consideration. Keywords—multi–criteria optimization, evolutionary algorithms, stochastic convergence to Pareto set