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

• Stũtzle T
• Dorigo M
• 83

• 314

Citations

Abstract

We prove some convergence properties for a class of ant colony
optimization algorithms. In particular, we prove that for any small
constant &#949; &gt; 0 and for a sufficiently large number of algorithm
iterations t, the probability of finding an optimal solution at least
once is P*(t) &ges; 1 - &#949; and that this probability tends to 1 for
t&rarr;&infin;. 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&rarr;&infin;, any fixed ant will produce
the optimal solution during the tth iteration with probability P &ges; 1
&#949;&circ;(&tau;min, &tau;max), where
&tau;min and &tau;max are the minimum and maximum
values that can be taken by pheromone trails

Author-supplied keywords

• ACO algorithms
• Ant algorithms
• Ant colony optimization
• Approximation algorithms
• Convergence proof
• Heuristics
• Metaheuristics

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