A rigorous runtime analysis of the 2-MMASib on jump functions: ant colony optimizers can cope well with local optima

Ant colony optimizers have been successfully used as general-purpose optimization heuristics. Due to the complicated nature of the random processes that describe the runs of ACO algorithms, the mathematical understanding of these algorithms is much less developed than that of other nature-inspired heuristics. In this first runtime analysis of a basic ACO algorithm on a classic multimodal benchmark, we analyze the runtime of the 2-MMASib on jump functions. For moderate jump sizes k ≤ α0 ln n, α0 > 0 a constant, we prove a runtime of order O(√n/ρ), when the evaporation factor ρ satisfies ρ ≤ Cn-1/2 ln(n)-1 for a sufficiently small constant C. For ρ = Θ(n-1/2 ln(n)-1), we thus obtain a runtime of O(n ln(n)). This result shows that simple ACO algorithms can cope much better with local optima than many evolutionary algorithms, which need Ω(nk) time.