A short convergence proof for a class of ant colony optimization algorithms
In this paper, we prove some convergence properties for a class of ant colony optimization algorithms. In particular, we prove that for any small constant epsilon greater than or equal to 0 and for a sufficiently large number of algorithm iterations t, the probability of finding an optimal solution at least once is P* (t) greater than or equal to 1 - epsilon and that this probability tends to 1 for t--> infinity. 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 -->infinity, any fixed ant will produce the optimal solution during the tth iteration with probability P greater than or equal to1 - is an element of((T)min, (T)max), where (T)min and(T)max are the minimum and maximum values that can be taken by pheromone trails..
ant colony optimization aco algorithms ant algorithms approximation algorithms convergence proof heuristics metaheuristics