Chapter 4: Ant Colony Optimization Theory

4.1 Theoretical Considerations on ACO

• Specifying the definition of ACO
• Convergence in value vs. Convergence in solution
• ACO bs, τmin
• Speed of convergence?

4.2 The Problem and the Algorighm

• Only considering a minimization problem (S, ƒ, Ω)
• Working only with static problems
• Best-so-far update is used

4.3 Convergence Proofs

Convergence in Value

• For ACO bs, τmin an optimal solution is guaranteed, but convergence in solution cannot be proved
• Proposition 4.1: The τmax is bound by the speed of evaporation
• Proposition 4.2: Once an optimal solution is found, the arcs of the solution will increase toward τmax
• Theorem 4.1: Since there is a non-zero probability of finding the optimum solution (s*), as the number of iterations tends toward infinity, the probability of finding s* becomes 1

Convergence in Solution

• For ACO bs, τmin(θ) convergence in solution is guaranteed, as is an optimum solution if τmin(θ) decreases at a very slow rate
• Theorem 4.2: if τmin(θ) = d / ln( θ+1 ), as the number of iterations increases the probability of finding s* goes to 1
• Proposition 4.3: Once s* is found, the arcs of the non-optimum solution go to 0
• Theorem 4.3: After finding s*, the probability of an ant constructing s* will reach 1 as θ reaches infinity

What Does the Proof Really Say?

• An Optimum solution is guaranteed for ACO bs, τmin and ACO bs, τmin(θ)
• No proofs have been done showing the time required to find s*
• In Theory, the smaller the ratio of τmax / τmin the greater the speed of convergence will be.
• In Practice, this often doesn't hold true
• These proofs can also hold for other ACO algorithms, such as MAX-MIN and ACS