Genetic algorithms with analytical solution

We study a class of genetic algorithms (GA) based on a population of chromosomes with mutation and crossover, as well as fitness. The chromosomes in these algorithms also have another property we call "activity" (which determines how fast a chromosome may mutate or interact with other chromosomes). We represent these algorithms by a multi-dimensional Markov process, and provide a new closed form expression for the probability distribution of the number of chromosomes of each type. Specifically we prove that in steady-state: (i) the joint steady-state probability of the number of chromosomes of each type is the product of the marginal probabilities for each type of chromosome, and (ii) the marginal probability of the number of chromosomes of each type has a geometric distribution. Another question we address is that of the choice of the GA parameters for a specific problem. We show that it is possible to choose parameters using a gradient based learning algorithm of complextity O(n3) and we apply this concept to the choice of the fitness parameter for our GA model.