On the convergence rates of genetic algorithms

He, Jifeng; Kang, Lishan

doi:10.1016/s0304-3975(99)00091-2

Cited by 88 publications

(57 citation statements)

References 13 publications

(26 reference statements)

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“…In such a case, the process (X t ; t = 0, 1, · · · ) can be modeled by a Markov chain [6], [19], [20], whose state space is the population space E, and the transition probability is…”

Section: A Evolutionary Algorithms and Markov Chain Modelsmentioning

confidence: 99%

From an individual to a population: an analysis of the first hitting time of population-based evolutionary algorithms

Yao

2002

IEEE Trans. Evol. Computat.

148

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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 population-based 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 population-based EAs will always be better than (1 + 1) EAs for all possible problems. [9], which make theoretical comparison between EAs and other optimization algorithms difficult. It is necessary to gain a deeper understanding of the time complexity of EAs in order to understand whether an EA is expected to scale well with the input size and when an EA is expected to provide the most benefits to a given problem. I. INTRODUCTION Evolutionary algorithms (EAsThere has been some work on the analysis of time complexity of (1 + 1) [8]. Because (1 + 1) EAs do not include recombination and population-based selection, the results on (1 + 1) EAs cannot be generalised to EAs with population size greater than one. It is important to understand the impact a population may have on an EA's average computation time. Such an understanding is expected to shed some light on the real utility of population-based EAs in combinatorial optimization [1], [2].In this paper, we compare (1 + 1) and (N + N ) EAs theoretically on two families of problems. We derive the first hitting time for (1 + 1) and (N + N ) EAs, respectively. Such results enable us to observe when the time would be polynomial or exponential in input size. The mathematical techniques used in this paper follow those in the analytical approach to the passage time of Markov chains [18]. Unlike drift analysis [8], which estimates the first hitting time from the drift of a Markov chain, these techniques calculate the first hitting time of a Markov chain directly from the transition matrix. The advantage over the drift analysis is that an exact expression of the first hitting time can be obtained for some EAs. The disadvantage is that such exact expressions are difficult, if not impossible, to derive from transition matrices if they are too complex.The rest of this paper is organized as follows. Section II introduces some notations, definitions and theorems about the first hitting time of a Markov chain. Section III contains our main results. Given typical problems, we deriv...

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“…In such a case, the process (X t ; t = 0, 1, · · · ) can be modeled by a Markov chain [6], [19], [20], whose state space is the population space E, and the transition probability is…”

Section: A Evolutionary Algorithms and Markov Chain Modelsmentioning

confidence: 99%

From an individual to a population: an analysis of the first hitting time of population-based evolutionary algorithms

Yao

2002

IEEE Trans. Evol. Computat.

148

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“…Since the state of ξ(t + 1) is only dependent on the state of ξ(t), then {ξ(t); t = 0, 1, · · · } can be modeled by a nonhomogeneous Markov chain whose state space is X and the transition function is [11]:…”

Section: Eas and Their Markov Chain Modelmentioning

confidence: 99%

“…In our previous paper [11], we have discussed the convergence rate of genetic algorithms from the viewpoint of abstract framework. In the same way, we will undertake further investigations into the sufficient conditions, necessary conditions, and sufficient and necessary conditions for the convergence of EAs, and their relationship.…”

Section: Introductionmentioning

confidence: 99%

Conditions for the convergence of evolutionary algorithms

2001

Journal of Systems Architecture

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This paper presents a theoretical analysis of the convergence conditions for evolutionary algorithms. The necessary and sufficient conditions, necessary conditions, and sufficient conditions for the convergence of evolutionary algorithms to the global optima are derived, which describe their limiting behaviors. Their relationships are explored. Upper and lower bounds of the convergence rates of the evolutionary algorithms are given.

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“…By using non-stationary Markov model, Cao and Wu (1997) investigated the asymptotic convergence property of the GA with adaptive crossover and mutation. Based on the theory about the convergence rate of Markov chain (Rosenthal 1995), He and Kang (1999) discussed the convergence rate of GA by using the minorization condition and obtained a bound on the convergence rate for the algorithm with time-invariant operators on a general state space as well as a bound on the convergence rate for the algorithm with time-variant operators on a finite state space. Based on Davis's work (1991), Suzuki (1998) discussed the theoretical property on the Markov chain of GA, which is applicable to SAlike strategies.…”

Section: Introductionmentioning

confidence: 99%

A unified framework for population-based metaheuristics

Liu

Wang

2011

Ann Oper Res

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Based on the analysis of the basic schemes of a variety of population-based metaheuristics (PBMH), the main components of PBMH are described with functional relationships in this paper and a unified framework (UF) is proposed for PBMH to provide a comprehensive way of viewing PBMH and to help understand the essential philosophy of PBMH from a systematic standpoint. The relevance of the proposed UF and some typical PBMH methods is illustrated, including particle swarm optimization, differential evolution, scatter search, ant colony optimization, genetic algorithm, evolutionary programming, and evolution strategies, which can be viewed as the instances of the UF. In addition, as a platform to develop the new population-based metaheuristics, the UF is further explained to provide some designing issues for effective/efficient PBMH algorithms. Subsequently, the UF is extended, namely UFmeme to describe the Memetic Algorithms (MAs) by adding local search (memetic component) to the framework as an extra-feature. Finally, we theoretically

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On the convergence rates of genetic algorithms

Cited by 88 publications

References 13 publications

From an individual to a population: an analysis of the first hitting time of population-based evolutionary algorithms

From an individual to a population: an analysis of the first hitting time of population-based evolutionary algorithms

Conditions for the convergence of evolutionary algorithms

A unified framework for population-based metaheuristics

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