In this paper we obtain bounds on the probability of convergence to the optimal solution for the compact genetic algorithm (cGA) and the population based incremental learning (PBIL). Moreover, we give a sufficient condition for convergence of these algorithms to the optimal solution and compute a range of possible values for algorithm parameters at which there is convergence to the optimal solution with a predefined confidence level.

The compact Genetic Algorithm (cGA) is an Estimation of Distribution Algorithm that generates offspring population according to the estimated probabilistic model of the parent population instead of using traditional recombination and mutation operators. The cGA only needs a small amount of memory; therefore, it may be quite useful in memory-constrained applications. This paper introduces a theoretical framework for studying the cGA from the convergence point of view in which, we model the cGA by a Markov process and approximate its behavior using an Ordinary Differential Equation (ODE). Then, we prove that the corresponding ODE converges to local optima and stays there. Consequently, we conclude that the cGA will converge to the local optima of the function to be optimized.