On the Finite-Time Dynamics of Ant Colony Optimization

An analytical framework for investigating the finite-time dynamics of ant colony optimization (ACO) under a fitness-proportional pheromone update rule on arbitrary construction graphs is developed. A limit theorem on the approximation of the stochastic ACO process by a deterministic process is demonstrated, and a system of ordinary differential equations governing the process dynamics is identified. As an example for the application of the presented theory, the behavior of ACO on three different construction graphs for subset selection problems is analyzed and compared for some basic test functions. The theory enables first rough theoretical predictions of the convergence speed of ACO.

[1]  Mikhail Borisovich Nevelʹson,et al.  Stochastic Approximation and Recursive Estimation , 1976 .

[2]  John H. Holland,et al.  When will a Genetic Algorithm Outperform Hill Climbing , 1993, NIPS.

[3]  Philippe Flajolet,et al.  An introduction to the analysis of algorithms , 1995 .

[4]  Rich Caruana,et al.  Removing the Genetics from the Standard Genetic Algorithm , 1995, ICML.

[5]  Magnus Rattray,et al.  Noisy Fitness Evaluation in Genetic Algorithms and the Dynamics of Learning , 1996, FOGA.

[6]  Marco Dorigo,et al.  Ant system: optimization by a colony of cooperating agents , 1996, IEEE Trans. Syst. Man Cybern. Part B.

[7]  B. Bullnheimer,et al.  A NEW RANK BASED VERSION OF THE ANT SYSTEM: A COMPUTATIONAL STUDY , 1997 .

[8]  A. Prügel-Bennett,et al.  The dynamics of a genetic algorithm for simple random Ising systems , 1997 .

[9]  Richard F. Hartl,et al.  An ant colony optimization approach for the single machine total tardiness problem , 1999, Proceedings of the 1999 Congress on Evolutionary Computation-CEC99 (Cat. No. 99TH8406).

[10]  E. D. Taillard,et al.  Ant Systems , 1999 .

[11]  Luca Maria Gambardella,et al.  Ant Algorithms for Discrete Optimization , 1999, Artificial Life.

[12]  Riccardo Poli,et al.  New ideas in optimization , 1999 .

[13]  Marco Dorigo,et al.  The ant colony optimization meta-heuristic , 1999 .

[14]  Walter J. Gutjahr,et al.  A Graph-based Ant System and its convergence , 2000, Future Gener. Comput. Syst..

[15]  Thomas Stützle,et al.  Proceedings of ANTS 2000 – From Ant Colonies to Artificial Ants: Second International Workshop on Ant Algorithms , 2000 .

[16]  Thomas Stützle,et al.  MAX-MIN Ant System , 2000, Future Gener. Comput. Syst..

[17]  T. Mahnig,et al.  Evolutionary algorithms: from recombination to search distributions , 2001 .

[18]  Richard F. Hartl,et al.  Ant Colony Optimization in Multiobjective Portfolio Selection , 2001 .

[19]  A. Prügel-Bennett,et al.  Modelling genetic algorithm dynamics , 2001 .

[20]  Martin Middendorf,et al.  Modeling the Dynamics of Ant Colony Optimization , 2002, Evolutionary Computation.

[21]  Thomas Stützle,et al.  A short convergence proof for a class of ant colony optimization algorithms , 2002, IEEE Trans. Evol. Comput..

[22]  Thomas Stützle,et al.  A SHORT CONVERGENCE PROOF FOR A CLASS OF ACO ALGORITHMS , 2002 .

[23]  Walter J. Gutjahr,et al.  ACO algorithms with guaranteed convergence to the optimal solution , 2002, Inf. Process. Lett..

[24]  Luca Maria Gambardella,et al.  Solving the Homogeneous Probabilistic Traveling Salesman Problem by the ACO Metaheuristic , 2002, Ant Algorithms.

[25]  Christine Solnon,et al.  Searching for Maximum Cliques with Ant Colony Optimization , 2003, EvoWorkshops.

[26]  Walter J. Gutjahr,et al.  A Converging ACO Algorithm for Stochastic Combinatorial Optimization , 2003, SAGA.

[27]  H. Kushner,et al.  Stochastic Approximation and Recursive Algorithms and Applications , 2003 .

[28]  W. Gutjahr.  A GENERALIZED CONVERGENCE RESULT FOR THE GRAPH-BASED ANT SYSTEM METAHEURISTIC , 2003, Probability in the Engineering and Informational Sciences.

[29]  Pat Langley,et al.  Editorial: On Machine Learning , 1986, Machine Learning.

[30]  Manuel López-Ibáñez,et al.  Ant colony optimization , 2010, GECCO '10.

[31]  W. Gutjahr.  S-ACO: An Ant-Based Approach to Combinatorial Optimization Under Uncertainty , 2004, ANTS Workshop.

[32]  Richard F. Hartl,et al.  Pareto Ant Colony Optimization: A Metaheuristic Approach to Multiobjective Portfolio Selection , 2004, Ann. Oper. Res..

[33]  Daniel Merkle,et al.  Competition Controlled Pheromone Update for Ant Colony Optimization , 2004, ANTS Workshop.

[34]  S. Brailsford,et al.  Optimal screening policies for diabetic retinopathy using a combined discrete-event simulation and ant colony optimization approach , 2005 .

[35]  Marco Dorigo,et al.  Search bias in ant colony optimization: on the role of competition-balanced systems , 2005, IEEE Transactions on Evolutionary Computation.

[36]  M. Dorigo,et al.  ACO/F-Race: Ant Colony Optimization and Racing Techniques for Combinatorial Optimization Under Uncertainty , 2005 .

[37]  Christian Blum,et al.  New metaheuristic approaches for the edge-weighted k-cardinality tree problem , 2005, Comput. Oper. Res..

[38]  Walter J. Gutjahr,et al.  An ACO algorithm for a dynamic regional nurse-scheduling problem in Austria , 2007, Comput. Oper. Res..

[39]  Walter J. Gutjahr,et al.  First steps to the runtime complexity analysis of ant colony optimization , 2008, Comput. Oper. Res..