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

  • Stũtzle T
  • Dorigo M
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Abstract

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(τmin, τmax), where
τmin and τmax 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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