Markov chain analysis of genetic algorithms applied to fitness functions perturbed concurrently by additive and multiplicative noise
We analyze the transition and convergence properties of genetic algorithms (GAs) applied to fitness functions perturbed concurrently by additive and multiplicative noise. Both additive noise and multiplicative noise are assumed to take on finitely many values. We explicitly construct a Markov chain...
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| Veröffentlicht in: | Computational optimization and applications Jg. 51; H. 2; S. 601 - 622 |
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| Abstract | We analyze the transition and convergence properties of genetic algorithms (GAs) applied to fitness functions perturbed concurrently by additive and multiplicative noise. Both additive noise and multiplicative noise are assumed to take on finitely many values. We explicitly construct a Markov chain that models the evolution of GAs in this noisy environment and analyze it to investigate the algorithms. Our analysis shows that this Markov chain is indecomposable; it has only one positive recurrent communication class. Using this property, we establish a condition that is both necessary and sufficient for GAs to eventually (i.e., as the number of iterations goes to infinity) find a globally optimal solution with probability 1. Similarly, we identify a condition that is both necessary and sufficient for the algorithms to eventually with probability 1 fail to find any globally optimal solution. Our analysis also shows that the chain has a stationary distribution that is also its steady-state distribution. Based on this property and the transition probabilities of the chain, we compute the exact probability that a GA is guaranteed to select a globally optimal solution upon completion of each iteration. |
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| AbstractList | We analyze the transition and convergence properties of genetic algorithms (GAs) applied to fitness functions perturbed concurrently by additive and multiplicative noise. Both additive noise and multiplicative noise are assumed to take on finitely many values. We explicitly construct a Markov chain that models the evolution of GAs in this noisy environment and analyze it to investigate the algorithms. Our analysis shows that this Markov chain is indecomposable; it has only one positive recurrent communication class. Using this property, we establish a condition that is both necessary and sufficient for GAs to eventually (i.e., as the number of iterations goes to infinity) find a globally optimal solution with probability 1. Similarly, we identify a condition that is both necessary and sufficient for the algorithms to eventually with probability 1 fail to find any globally optimal solution. Our analysis also shows that the chain has a stationary distribution that is also its steady-state distribution. Based on this property and the transition probabilities of the chain, we compute the exact probability that a GA is guaranteed to select a globally optimal solution upon completion of each iteration. We analyze the transition and convergence properties of genetic algorithms (GAs) applied to fitness functions perturbed concurrently by additive and multiplicative noise. Both additive noise and multiplicative noise are assumed to take on finitely many values. We explicitly construct a Markov chain that models the evolution of GAs in this noisy environment and analyze it to investigate the algorithms. Our analysis shows that this Markov chain is indecomposable; it has only one positive recurrent communication class. Using this property, we establish a condition that is both necessary and sufficient for GAs to eventually (i.e., as the number of iterations goes to infinity) find a globally optimal solution with probability 1. Similarly, we identify a condition that is both necessary and sufficient for the algorithms to eventually with probability 1 fail to find any globally optimal solution. Our analysis also shows that the chain has a stationary distribution that is also its steady-state distribution. Based on this property and the transition probabilities of the chain, we compute the exact probability that a GA is guaranteed to select a globally optimal solution upon completion of each iteration. We analyze the transition and convergence properties of genetic algorithms (GAs) applied to fitness functions perturbed concurrently by additive and multiplicative noise. Both additive noise and multiplicative noise are assumed to take on finitely many values. We explicitly construct a Markov chain that models the evolution of GAs in this noisy environment and analyze it to investigate the algorithms. Our analysis shows that this Markov chain is indecomposable; it has only one positive recurrent communication class. Using this property, we establish a condition that is both necessary and sufficient for GAs to eventually (i.e., as the number of iterations goes to infinity) find a globally optimal solution with probability 1. Similarly, we identify a condition that is both necessary and sufficient for the algorithms to eventually with probability 1 fail to find any globally optimal solution. Our analysis also shows that the chain has a stationary distribution that is also its steady-state distribution. Based on this property and the transition probabilities of the chain, we compute the exact probability that a GA is guaranteed to select a globally optimal solution upon completion of each iteration.[PUBLICATION ABSTRACT] |
| Author | Nakama, Takehiko |
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| Cites_doi | 10.1109/TEVC.2005.846356 10.1109/4235.735433 10.1109/4235.910461 10.1145/1389095.1389283 10.1109/21.370197 10.1016/S0045-7825(99)00386-2 10.1109/72.265964 10.1007/s11081-006-9970-y 10.1007/BF01530778 10.1109/ACC.2007.4282251 10.1162/evco.1993.1.3.269 10.1137/0325087 10.1162/evco.1996.4.2.113 10.1007/978-3-540-70560-4_11 10.7551/mitpress/6229.001.0001 10.1007/978-1-4615-1105-2 |
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| Keywords | Markov chain analysis Evolutionary computation Perturbed fitness functions Additive and multiplicative noise Noisy environments Genetic algorithms Convergence analysis |
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| References_xml | – reference: VoseM.D.LiepinsG.E.Punctuated equilibria in genetic searchComplex Syst.19915314411164200764.68149 – reference: GoldbergD.E.RudnickM.W.Genetic algorithms and the variance of fitnessComplex Syst.199152652780729.68075 – reference: ChenA.SubprasomK.JiZ.A simulation-based multi-objective genetic algorithm (SMOGA) procedure for BOT network design problemOptim. Eng.2006722524722701521138.9046610.1007/s11081-006-9970-y – reference: JebaliaM.AugerA.On multiplicative noise models for stochastic searchProceedings of the 2008 International Conference on Parallel Problem Solving from Nature20085261 – reference: KarlinS.TaylorH.M.A First Course in Stochastic Processes1975New YorkAcademic Press0315.60016 – reference: NakamaT.Markov chain analysis of genetic algorithms applied to fitness functions perturbed by multiple sources of additive noiseStud. Comput. Intell.200814912313610.1007/978-3-540-70560-4_11 – reference: DavisT.E.PrincipeJ.A Markov chain framework for the simple genetic algorithmEvol. Comput.1993126928810.1162/evco.1993.1.3.269 – reference: Di PietroA.WhiteL.BaroneL.Applying evolutionary algorithms to problems with noisy, time-consuming fitness functionsProceedings of the 2004 Congress on Evolutionary Computation200412541261 – reference: ArnoldD.V.Noisy Optimization with Evolution Strategies2002BostonKluwer Academic1103.9000510.1007/978-1-4615-1105-2 – reference: BeyerH.G.Evolutionary algorithms in noisy environments: theoretical issues and guidelines for practiceComput. Methods Appl. Mech. Eng.20001862392671064.9057410.1016/S0045-7825(99)00386-2 – reference: NakamaT.Theoretical analysis of genetic algorithms in noisy environments based on a Markov modelProceedings of the 2008 Genetic and Evolutionary Computation Conference20081001100810.1145/1389095.1389283 – reference: VoseM.D.The Simple Genetic Algorithm1999CambridgeMIT Press0952.65048 – reference: LeungK.S.DuanQ.H.XuZ.B.WongC.K.A new model of simulated evolutionary computation—convergence analysis and specificationsIEEE Trans. Evol. Comput.2001531610.1109/4235.910461 – reference: PrimbsJ.A.Portfolio optimization applications of stochastic receding horizon controlProceedings of the 2007 American Control Conference20071811181610.1109/ACC.2007.4282251 – reference: RudolphG.Partial order approach to noisy fitness functionsProceedings of the 2001 IEEE Congress on Evolutionary Computation2001318325 – reference: NixA.VoseM.D.Modeling genetic algorithm with Markov chainsAnn. Math. Artif. Intell.199252734127941710.1007/BF01530781 – reference: SuzukiJ.A Markov chain analysis on simple genetic algorithmsIEEE Trans. Syst. Man Cybern.19952565565910.1109/21.370197 – reference: NissenV.PropachJ.On the robustness of population-based versus point-based optimization in the presence of noiseIEEE Trans. Evol. Comput.1998210711910.1109/4235.735433 – reference: HopkinsW.E.Optimal stabilization of families of linear stochastic differential equations with jump coefficients and multiplicative noiseSIAM J. Control Optim.198725158716009124570636.9307710.1137/0325087 – reference: MillerB.L.GoldbergD.E.Genetic algorithms, selection schemes, and the varying effects of noiseEvol. Comput.1996411313110.1162/evco.1996.4.2.113 – reference: RudolphG.Convergence analysis of canonical genetic algorithmsIEEE Trans. Neural Netw.199459610110.1109/72.265964 – reference: RudinL.I.OsherS.Total variation based image restoration with free local constraintsProceedings of the 2004 International Conference on Image Processing20043135 – reference: JinY.BrankeJ.Evolutionary optimization in uncertain environments—a surveyIEEE Trans. Evol. Comput.2005330331710.1109/TEVC.2005.846356 – reference: KarlinS.TaylorH.M.A Second Course in Stochastic Processes1981New YorkAcademic Press0469.60001 – volume: 5 start-page: 31 year: 1991 ident: 9371_CR24 publication-title: Complex Syst. – volume-title: A Second Course in Stochastic Processes year: 1981 ident: 9371_CR11 – volume: 3 start-page: 303 year: 2005 ident: 9371_CR9 publication-title: IEEE Trans. Evol. 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| SubjectTerms | Additives Algorithms Chains Chromosomes Convex and Discrete Geometry Elitism Expected values Genetic algorithms Investigations Management Science Markov analysis Markov chains Mathematical models Mathematics Mathematics and Statistics Mutation Noise Operations Research Operations Research/Decision Theory Optimization Population Probability Statistics Studies |
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| Title | Markov chain analysis of genetic algorithms applied to fitness functions perturbed concurrently by additive and multiplicative noise |
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