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24
Runtime Analysis of a Simple Ant Colony Optimization Algorithm
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 84 (2006)
, 2006
"... Ant Colony Optimization (ACO) has become quite popular in recent years. In contrast to many successful applications, the theoretical foundation of this randomized search heuristic is rather weak. Building up such a theory is demanded to understand how these heuristics work as well as to come up with ..."
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Cited by 49 (13 self)
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Ant Colony Optimization (ACO) has become quite popular in recent years. In contrast to many successful applications, the theoretical foundation of this randomized search heuristic is rather weak. Building up such a theory is demanded to understand how these heuristics work as well as to come up with better algorithms for certain problems. Up to now, only convergence results have been achieved showing that optimal solutions can be obtained in finite time. We present the first runtime analysis of an ACO algorithm, which transfers many rigorous results with respect to the runtime of a simple evolutionary algorithm to our algorithm. Moreover, we examine the choice of the evaporation factor, a crucial parameter in ACO algorithms, in detail. By deriving new lower bounds on the tails of sums of independent Poisson trials, we determine the effect of the evaporation factor almost completely and prove a phase transition from exponential to polynomial runtime.
A graphbased Ant System and its convergence
 Future Generation Computing Systems
, 2000
"... Abstract: A general framework for solving combinatorial optimization problems heuristically by the Ant System approach is developed. The framework is based on the concept of a construction graph, a graph assigned to an instance of the optimization problem under consideration, encoding feasible solut ..."
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Cited by 35 (1 self)
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Abstract: A general framework for solving combinatorial optimization problems heuristically by the Ant System approach is developed. The framework is based on the concept of a construction graph, a graph assigned to an instance of the optimization problem under consideration, encoding feasible solutions by walks. It is shown that under certain conditions, the solutions generated in each iteration of this Graph–based Ant System converge with a probability that can be made arbitrarily close to one to the optimal solution of the given problem instance.
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 32 (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.
Application of the CrossEntropy Method to the Buffer Allocation Problem in a SimulationBased Environment
 Annals of Operations Research
, 2005
"... Abstract. The buffer allocation problem (BAP) is a wellknown difficult problem in the design of production lines. We present a stochastic algorithm for solving the BAP, based on the crossentropy method,anew paradigm for stochastic optimization. The algorithm involves the following iterative steps: ..."
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Cited by 21 (4 self)
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Abstract. The buffer allocation problem (BAP) is a wellknown difficult problem in the design of production lines. We present a stochastic algorithm for solving the BAP, based on the crossentropy method,anew paradigm for stochastic optimization. The algorithm involves the following iterative steps: (a) the generation of buffer allocations according to a certain random mechanism, followed by (b) the modification of this mechanism on the basis of crossentropy minimization. Through various numerical experiments we demonstrate the efficiency of the proposed algorithm and show that the method can quickly generate (near)optimal buffer allocations for fairly large production lines.
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.
On the runtime analysis of the 1ANT ACO algorithm
 IN GENETIC AND EVOLUTIONARY COMPUTATION CONFERENCE, GECCO 2007, PROCEEDINGS
, 2007
"... The runtime analysis of randomized search heuristics is a growing field where, in the last two decades, many rigorous results have been obtained. These results, however, apply particularly to classical search heuristics such as Evolutionary Algorithms (EAs) and Simulated Annealing. First runtime ana ..."
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Cited by 17 (11 self)
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The runtime analysis of randomized search heuristics is a growing field where, in the last two decades, many rigorous results have been obtained. These results, however, apply particularly to classical search heuristics such as Evolutionary Algorithms (EAs) and Simulated Annealing. First runtime analyses of modern search heuristics have been conducted only recently w. r. t. a simple Ant Colony Optimization (ACO) algorithm called 1ANT. In particular, the influence of the evaporation factor in the pheromone update mechanism and the robustness of this parameter w. r. t. the runtime behavior have been determined for the example function OneMax. This paper puts forward the rigorous runtime analysis of the 1ANT on example functions, namely on the functions LeadingOnes and BinVal. With respect to EAs, such analyses have been essential to develop methods for the analysis on more complicated problems. The proof techniques required for the 1ANT, unfortunately, differ significantly from those for EAs, which means that a new reservoir of methods has to be built up. Again, the influence of the evaporation factor is analyzed rigorously, and it is proved that its choice can be very crucial to allow efficient runtimes. Moreover, the analyses provide insight into the working principles of ACO algorithms and, in terms of their robustness, describe essential differences to other randomized search heuristics.
Refined runtime analysis of a basic ant colony optimization algorithm
 In IEEE Congress on Evolutionary Computation 2007
, 2007
"... Neumann and Witt (2006) analyzed the runtime of the basic ant colony optimization (ACO) algorithm 1Ant on pseudoboolean optimization problems. For the problem OneMax they showed how the runtime depends on the evaporation factor. In particular, they proved a phase transition from exponential to poly ..."
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Cited by 11 (1 self)
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Neumann and Witt (2006) analyzed the runtime of the basic ant colony optimization (ACO) algorithm 1Ant on pseudoboolean optimization problems. For the problem OneMax they showed how the runtime depends on the evaporation factor. In particular, they proved a phase transition from exponential to polynomial runtime. In this work, we simplify the view on this problem by an appropriate translation of the pheromone model. This results in a profound simplification of the pheromone update rule and, by that, a refinement of the results of Neumann and Witt. In particular, we show how the exponential runtime bound gradually changes to a polynomial bound inside the phase of transition. 1
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.
Application of the Cross Entropy Method for Buffer Allocation Problem in Simulation Based Environment
 In preparation
"... This paper deals with application of the crossentropy (CE) method for solving the buffer allocation problem (BAP), i.e., with determination of optimal buffer allocation in a serial production line with the objective of maximizing the throughput. The crossentropy method for BAP presents an adaptive ..."
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Cited by 7 (0 self)
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This paper deals with application of the crossentropy (CE) method for solving the buffer allocation problem (BAP), i.e., with determination of optimal buffer allocation in a serial production line with the objective of maximizing the throughput. The crossentropy method for BAP presents an adaptive algorithm equipped with an auxiliary random mechanism, which transforms the original deterministic problem into an associated stochastic one. The random mechanism in BAP presents an (m 1) x (n + 1) probability matrix P = (P ij), such that P n j=0 P ij = 1; 8i = 1; : : : ; m 1, where m 1 and n denote the number of niches (the number of machines is the number of niches +1) and the buer spaces, respectively, and of the P ijth element of P represents the probability of niche i receiving j buer spaces (note that allocation of 0 buers in some niche is feasible). Once the matrix P is dened, each iteration of the CE algorithm comprises of the following two phases: