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On the Analysis of the (1+1) Evolutionary Algorithm
 THEORETICAL COMPUTER SCIENCE
, 2002
"... Many experimental results are reported on all types of Evolutionary Algorithms but only few results have been proved. A step towards a theory on Evolutionary Algorithms, in particular, the socalled (1 + 1) Evolutionary Algorithm, is performed. Linear functions are proved to be optimized in expected ..."
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Cited by 225 (36 self)
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Many experimental results are reported on all types of Evolutionary Algorithms but only few results have been proved. A step towards a theory on Evolutionary Algorithms, in particular, the socalled (1 + 1) Evolutionary Algorithm, is performed. Linear functions are proved to be optimized in expected time O(n ln n) but only mutation rates of size #(1/n) can ensure this behavior. For some polynomial of degree 2 the optimization needs exponential time. The same is proved for a unimodal function. Both results were not expected by several other authors. Finally, a hierarchy result is proved. Moreover, methods are presented to analyze the behavior of the (1 + 1) Evolutionary Algorithm.
From an individual to a population: An analysis of the first hitting time of populationbased evolutionary algorithms
 IEEE Transactions on Evolutionary Computation
, 2002
"... Almost all analyses of time complexity of evolutionary algorithms (EAs) have been conducted for (1+1) EAs only. Theoretical results on the average computation time of populationbased EAs are few. However, the vast majority of applications of EAs use a population size that is greater than one. The u ..."
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Cited by 57 (17 self)
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Almost all analyses of time complexity of evolutionary algorithms (EAs) have been conducted for (1+1) EAs only. Theoretical results on the average computation time of populationbased EAs are few. However, the vast majority of applications of EAs use a population size that is greater than one. The use of population has been regarded as one of the key features of EAs. It is important to understand in depth what the real utility of population is in terms of the time complexity of EAs, when EAs are applied to combinatorial optimization problems. This paper compares (1 + 1) EAs and (N + N) EAs theoretically by deriving their first hitting time on the same problems. It is shown that a population can have a drastic impact on an EA’s average computation time, changing an exponential time to a polynomial time (in the input size) in some cases. It is also shown that the first hitting probability can be improved by introducing a population. However, the results presented in this paper do not imply that populationbased EAs will always be better than (1 + 1) EAs for all possible problems. I.
Towards an analytic framework for analysing the computation time of evolutionary algorithms
 Artificial Intelligence
, 2003
"... In spite of many applications of evolutionary algorithms in optimisation, theoretical results on the computation time and time complexity of evolutionary algorithms on different optimisation problems are relatively few. It is still unclear when an evolutionary algorithm is expected to solve an optim ..."
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Cited by 54 (18 self)
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In spite of many applications of evolutionary algorithms in optimisation, theoretical results on the computation time and time complexity of evolutionary algorithms on different optimisation problems are relatively few. It is still unclear when an evolutionary algorithm is expected to solve an optimisation problem efficiently or otherwise. This paper gives a general analytic framework for analysing first hitting times of evolutionary algorithms. The framework is built on the absorbing Markov chain model of evolutionary algorithms. The first step towards a systematic comparative study among different EAs and their first hitting times has been made in the paper.
Evolutionary Algorithms  How to Cope With Plateaus of Constant Fitness and When to Reject Strings of The Same Fitness
, 2000
"... The most simple evolutionary algorithm, the socalled (1+1)EA accepts a child if its fitness is at least as large (in the case of maximization) as the fitness of its parent. The variant (1 + 1) # EA only accepts a child if its fitness is strictly larger than the fitness of its parent. Here two funct ..."
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Cited by 53 (12 self)
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The most simple evolutionary algorithm, the socalled (1+1)EA accepts a child if its fitness is at least as large (in the case of maximization) as the fitness of its parent. The variant (1 + 1) # EA only accepts a child if its fitness is strictly larger than the fitness of its parent. Here two functions related to the class of long path functions are presented such that the (1 + 1)EA maximizes one of it in polynomial time and needs exponential time for the other while the (1+1) # EA has the opposite behavior. These results prove that small changes of an evolutionary algorithm may change its behavior significantly. Since the (1 + 1)EA and the (1 + 1) # EA di#er only on plateaus of constant fitness, the results also show how evolutionary algorithms behave on such plateaus. The (1 + 1)EA can pass a path of constant fitness and polynomial length in polynomial time. Finally, for these functions it is shown that local performance measures like the quality gain and the progress rate do not de...
Running Time Analysis of a MultiObjective Evolutionary Algorithm on a Simple Discrete Optimization Problem
, 2002
"... For the first time, a running time analysis of a multiobjective evolutionary algorithm for a discrete optimization problem is given. To this end, a simple pseudoBoolean problem (Lotz: leading ones  trailing zeroes) is defined and a populationbased optimization algorithm (FEMO). We show, that the ..."
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Cited by 53 (8 self)
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For the first time, a running time analysis of a multiobjective evolutionary algorithm for a discrete optimization problem is given. To this end, a simple pseudoBoolean problem (Lotz: leading ones  trailing zeroes) is defined and a populationbased optimization algorithm (FEMO). We show, that the algorithm performs a black box optimization in #(n 2 log n) function evaluations where n is the number of binary decision variables. 1
An Analysis of the (µ+1) EA on Simple PseudoBoolean Functions (Extended Abstract)
, 2004
"... Evolutionary Algorithms (EAs) are successfully applied for optimization in discrete search spaces, but theory is still weak in particular for populationbased EAs. Here, a first rigorous analysis of the (+1) EA on pseudoBoolean functions is presented. For three example functions wellknown fr ..."
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Cited by 44 (7 self)
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Evolutionary Algorithms (EAs) are successfully applied for optimization in discrete search spaces, but theory is still weak in particular for populationbased EAs. Here, a first rigorous analysis of the (+1) EA on pseudoBoolean functions is presented. For three example functions wellknown from the analysis of the (1+1) EA, bounds on the expected runtime and success probability are derived. For two of these functions, upper and lower bounds on the expected runtime are tight, and the (+1) EA is never more efficient than the (1+1) EA. Moreover, all lower bounds grow with . On a more complicated function, however, a small increase of provably decreases the expected runtime drastically. For the lower bounds,
Theoretical Aspects of Evolutionary Algorithms
, 2001
"... Randomized search heuristics like simulated annealing and evolutionary algorithms are applied successfully in many different situations. However, the theory on these algorithms is still in its infancy. Here it is discussed how and why such a theory should be developed. Afterwards, some fundament ..."
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Cited by 32 (12 self)
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Randomized search heuristics like simulated annealing and evolutionary algorithms are applied successfully in many different situations. However, the theory on these algorithms is still in its infancy. Here it is discussed how and why such a theory should be developed. Afterwards, some fundamental results on evolutionary algorithms are presented in order to show how theoretical results on randomized search heuristics can be proved and how they contribute to the understanding of evolutionary algorithms.
How To Analyse Evolutionary Algorithms
, 2002
"... Many variants of evolutionary algorithms have been designed and applied. The ..."
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Cited by 31 (1 self)
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Many variants of evolutionary algorithms have been designed and applied. The
A New Framework for the Valuation of Algorithms for BlackBoxOptimization
, 2001
"... Blackbox optimization algorithms cannot use the specific parameters of the problem instance, i.e., of the fitness function f. Their run time is measured as the number of fevaluations. This implies that the usual algorithmic complexity of a problem cannot be used in the blackbox scenario. Therefor ..."
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Cited by 28 (11 self)
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Blackbox optimization algorithms cannot use the specific parameters of the problem instance, i.e., of the fitness function f. Their run time is measured as the number of fevaluations. This implies that the usual algorithmic complexity of a problem cannot be used in the blackbox scenario. Therefore, a new framework for the valuation of algorithms for blackbox optimization is presented allowing the notion of the blackbox complexity of a problem. For several problems upper and lower bounds on their blackbox complexity are presented. Moreover, it can can be concluded that randomized search heuristics whose (worstcase) expected optimization time for some problem is close to the blackbox complexity of the problem are provably efficient (in the blackbox scenario). The new approach is applied to several problems based on typical example functions and further interesting problems. Run times of general EAs for these problems are compared with the blackbox complexity of the problem.