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On the BMDGAs and Neural Nets

We analyze a bivariate marginal distribution genetic model in case of infinite populations and provide relations between the associated infinite population genetic system and the neural networks. A lower bound on population size is exhibited stating that the behaviour of the finite population system, in case of sufficiently large sizes, can be suitably approximated by the behaviour of the corresponding infinite population system for a number of transitions exponentially greater than that suggested by Vose's analysis. The infinite population system is analyzed by showing that, conversely to what happens in the univariate case, the fitness is not a Lyapunov function for its asynchronous variant. The attractors (with binary components) of the infinite population genetic system are characterized as equilibrium points of a discrete (neural network) system that can be considered as a variant of a Hopfield's network; it is shown that the fitness is a Lyapunov function for the variant of the discrete Hopfield's net. The genetic algorithm based on the proposed infinite population system is experimentally compared with the (neural) network algorithm for the max-cut problem. Our main result can be summarized by stating that the relation between marginal distribution genetic systems and neural nets is much more general than that already shown elsewhere for the univariate models.