A short convergence proof for a class of ant colony optimization algorithms

We prove some convergence properties for a class of ant colony optimization algorithms. In particular, we prove that for any small constant ε > 0 and for a sufficiently large number of algorithm iterations t, the probability of finding an optimal solution at least once is P*(t) ⩾ 1 - ε and that this probability tends to 1 for t→∞. We also prove that, after an optimal solution has been found, it takes a finite number of iterations for the pheromone trails associated to the found optimal solution to grow higher than any other pheromone trail and that, for t→∞, any fixed ant will produce the optimal solution during the tth iteration with probability P ⩾ 1 εˆ(τ_min, τ_max), where τ_min and τ_max are the minimum and maximum values that can be taken by pheromone trails