A Pheromone-Rate-Based Analysis on the Convergence Time of ACO Algorithm

• Published in: IEEE transactions on systems, man and cybernetics. Part B, Cybernetics Vol. 39; no. 4; pp. 910 - 923
• Main Authors: Huang, Han, Wu, Chun-Guo, Hao, Zhi-Feng
• Format: Journal Article
• Language: English
• Published: United States IEEE 01.08.2009
• Subjects: Algorithm design and analysis; Algorithms; Animals; Ant colony optimization; Ant colony optimization (ACO); Ants - physiology; Artificial intelligence; Biological system modeling; Combinatorial analysis; Computer science; Convergence; Convergence of numerical methods; convergence time; Cybernetics; Cybernetics - methods; Deviation; Iterative methods; Laboratories; Markov Chains; Mathematical models; Models, Biological; Models, Statistical; nature-inspired cybernetic mechanisms; Optimization; pheromone rate; Pheromones - physiology; Runtime; runtime analysis
• ISSN: 1083-4419; 1941-0492; 1941-0492
• DOI: 10.1109/TSMCB.2009.2012867

Ant colony optimization (ACO) has widely been applied to solve combinatorial optimization problems in recent years. There are few studies, however, on its convergence time, which reflects how many iteration times ACO algorithms spend in converging to the optimal solution. Based on the absorbing Markov chain model, we analyze the ACO convergence time in this paper. First, we present a general result for the estimation of convergence time to reveal the relationship between convergence time and pheromone rate. This general result is then extended to a two-step analysis of the convergence time, which includes the following: 1) the iteration time that the pheromone rate spends on reaching the objective value and 2) the convergence time that is calculated with the objective pheromone rate in expectation. Furthermore, four brief ACO algorithms are investigated by using the proposed theoretical results as case studies. Finally, the conclusions of the case studies that the pheromone rate and its deviation determine the expected convergence time are numerically verified with the experiment results of four one-ant ACO algorithms and four ten-ant ACO algorithms.