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27
Ant Colony Optimization  Artificial Ants as a Computational Intelligence Technique
 IEEE COMPUT. INTELL. MAG
, 2006
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A short convergence proof for a class of Ant Colony Optimization algorithms
 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION
, 2002
"... In this paper, we prove some convergence properties for a class of ant colony optimization algorithms. In particular, we prove that for any small constant 0 and for a sufficiently large number of algorithm iterations, the probability of finding an optimal solution at least once is ( ) 1 and that th ..."
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Cited by 33 (1 self)
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In this paper, we prove some convergence properties for a class of ant colony optimization algorithms. In particular, we prove that for any small constant 0 and for a sufficiently large number of algorithm iterations, the probability of finding an optimal solution at least once is ( ) 1 and that this probability tends to 1 for. 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, any fixed ant will produce the optimal solution during the th iteration with probability 1 ^ ( min max), where min and max are the minimum and maximum values that can be taken by pheromone trails.
A Review on the Ant Colony Optimization Metaheuristic: Basis, Models and New Trends
 Mathware & Soft Computing
, 2002
"... Ant Colony Optimization (ACO) is a recent metaheuristic method that is inspired by the behavior of real ant colonies. In this paper, we review the underlying ideas of this approach that lead from the biological inspiration to the ACO metaheuristic, which gives a set of rules of how to apply ACO ..."
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Cited by 30 (2 self)
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Ant Colony Optimization (ACO) is a recent metaheuristic method that is inspired by the behavior of real ant colonies. In this paper, we review the underlying ideas of this approach that lead from the biological inspiration to the ACO metaheuristic, which gives a set of rules of how to apply ACO algorithms to challenging combinatorial problems. We present some of the algorithms that were developed under this framework, give an overview of current applications, and analyze the relationship between ACO and some of the best known metaheuristics. In addition, we describe recent theoretical developments in the eld and we conclude by showing several new trends and new research directions in this eld.
A Generalized Convergence Result for the GraphBased ant System MetaHeuristic
"... Abstract: It is shown that on fairly weak conditions, the current solutions of a metaheuristic following the ant colony optimization paradigm, the Graph{Based Ant System, converge with a probability that can be made arbitrarily close to unity to one element of the set of optimal solutions. The resul ..."
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Cited by 24 (5 self)
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Abstract: It is shown that on fairly weak conditions, the current solutions of a metaheuristic following the ant colony optimization paradigm, the Graph{Based Ant System, converge with a probability that can be made arbitrarily close to unity to one element of the set of optimal solutions. The result generalizes a previous result by removing the very restrictive condition that both the optimal solution and its encoding are unique (this generalization makes the proof distinctly more di±cult), and by allowing a wide class of implementation variants in the ¯rst phase of the algorithm. In this way, the range of application of the convergence result is considerably extended. 1
First steps to the runtime complexity analysis of Ant Colony Optimization
 Comput. Oper. Res
"... Abstract: The paper presents results on the runtime complexity of two ant colony optimization (ACO) algorithms: Ant System, the oldest ACO variant, and GBAS, the first ACO variant for which theoretical convergence results have been established. In both cases, as the class of test problems under cons ..."
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Cited by 20 (1 self)
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Abstract: The paper presents results on the runtime complexity of two ant colony optimization (ACO) algorithms: Ant System, the oldest ACO variant, and GBAS, the first ACO variant for which theoretical convergence results have been established. In both cases, as the class of test problems under consideration, a slight generalization of the wellknown OneMax test function has been chosen. The techniques used for the runtime analysis of the two algorithms differ: In the case of GBAS, the expected runtime until the optimal solution is reached is studied by a direct bound estimation approach inspired by comparable results for the (1+1) evolutionary algorithm (EA). A runtime bound of order O(m log m), where m is the problem instance size, is obtained. In the case of Ant System, the original discrete stochastic process is approximated by a suitable continuous deterministic process. The validity of the approximation is shown by means of a rigid convergence theorem exploiting a classical result from mathematical learning theory. Using this approximation, it is demonstrated that for the considered OneMaxtype problems, a runtime of order O(m log(1/ɛ)) until reaching an expected relative solution quality of 1 − ɛ, and a runtime of O(m log m) until reaching the optimal solution with high probability can be predicted. Our results are the first to show competitiveness in runtime complexity with (1+1) EA on OneMax for a proper ACO algorithm. 1
A converging ACO algorithm for stochastic combinatorial optimization
 Proc. SAGA 2003 Stochastic Algorithms: Foundations and Applications
, 2003
"... Abstract. The paper presents a generalpurpose algorithm for solving stochastic combinatorial optimization problems with the expected value of a random variable as objective and deterministic constraints. The algorithm follows the Ant Colony Optimization (ACO) approach and uses MonteCarlo sampling ..."
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Cited by 18 (5 self)
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Abstract. The paper presents a generalpurpose algorithm for solving stochastic combinatorial optimization problems with the expected value of a random variable as objective and deterministic constraints. The algorithm follows the Ant Colony Optimization (ACO) approach and uses MonteCarlo sampling for estimating the objective. It is shown that on rather mild conditions, including that of linear increment of the sample size, the algorithm converges with probability one to the globally optimal solution of the stochastic combinatorial optimization problem. Contrary to most convergence results for metaheuristics in the deterministic case, the algorithm can usually be recommended for practical application in an unchanged form, i.e., with the “theoretical ” parameter schedule.
SACO: An antbased approach to combinatorial optimization under uncertainty
 Proc. ANTS 2004 (4th International Workshop on Ant Colony Optimization and Swarm Intelligence), Springer LNCS 3172
, 2004
"... Abstract. A generalpurpose, simulationbased algorithm SACO for solving stochastic combinatorial optimization problems by means of the ant colony optimization (ACO) paradigm is investigated. Whereas in a prior publication, theoretical convergence of SACO to the globally optimal solution has been ..."
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Cited by 18 (4 self)
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Abstract. A generalpurpose, simulationbased algorithm SACO for solving stochastic combinatorial optimization problems by means of the ant colony optimization (ACO) paradigm is investigated. Whereas in a prior publication, theoretical convergence of SACO to the globally optimal solution has been demonstrated, the present article is concerned with an experimental study of SACO on two stochastic problems of fixedroutes type: First, a pretest is carried out on the probabilistic traveling salesman problem. Then, more comprehensive tests are performed for a traveling salesman problem with time windows (TSPTW) in the case of stochastic service times. As a yardstick, a stochastic simulated annealing (SSA) algorithm has been implemented for comparison. Both approaches are tested at randomly generated problem instances of different size. It turns out that SACO outperforms the SSA approach on the considered test instances. Some conclusions for finetuning SACO are drawn. 1
Ant Colony Optimization
 OPTIMIZATION TECHNIQUES IN ENGINEERING. SPRINGERVERLAG
, 2004
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On the finitetime dynamics of ant colony optimization
 Methodol. Comput. Appl. Probab
"... Abstract. An analytical framework for investigating the finitetime dynamics of ant colony optimization (ACO) under a fitnessproportional pheromone update rule on arbitrary construction graphs is developed. A limit theorem on the approximation of the stochastic ACO process by a deterministic proces ..."
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Cited by 9 (1 self)
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Abstract. An analytical framework for investigating the finitetime dynamics of ant colony optimization (ACO) under a fitnessproportional 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.
Mathematical runtime analysis of aco algorithms: survey on an emerging issue
 Swarm Intelligence
, 1999
"... Abstract: The paper gives an overview on the status of the theoretical analysis of Ant Colony Optimization (ACO) algorithms, with a special focus on the analytical investigation of the runtime required to find an optimal solution to a given combinatorial optimization problem. First, a general framew ..."
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Cited by 8 (0 self)
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Abstract: The paper gives an overview on the status of the theoretical analysis of Ant Colony Optimization (ACO) algorithms, with a special focus on the analytical investigation of the runtime required to find an optimal solution to a given combinatorial optimization problem. First, a general framework for studying questions of this type is presented, and three important ACO variants are recalled within this framework. Secondly, two classes of formal techniques for runtime investigations of the considered type are outlined. Finally, some available runtime complexity results for ACO variants, referring to elementary test problems that have been introduced in the theoretical literature on evolutionary algorithms, are cited and discussed. 1