遗传算法求解多模态优化问题的研究
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摘要
如何应用遗传算法求解多模态优化问题已经成为遗传算法乃至进化计算领域的一个广受关注的问题,它也是遗传算法理论和实际应用的基础。这方面的成果层出不穷,但理论研究相对薄弱。特别是研究外部参数如何影响遗传算法求解多模态优化问题的性能这方面仍然是个空白,本文着重进行了这方面的理论研究并对相应结论进行了推广。推广后的理论结果被用于改进遗传算法对多模态优化问题的搜索性能。本文的主要研究内容包括:
1.对多模态优化问题和遗传算法进行了简单回顾,对应用遗传算法解决多模态优化问题的研究现状进行了分析,介绍了遗传算法的基础理论及其发展趋势。
2.对经Walsh变换后的基因池遗传算法的无限种群模型进行了分析,刻画了“双峰函数”局部极值解的适值差与基因池遗传算法的搜索性能之间的关系,对上述理论结果进行了理论推广和实验验证。最后,根据所得结论设计出了针对多模态优化问题的两阶段遗传算法,并得到了满意的实验结果,这也从侧面说明了上述理论结果的正确性。
3.讨论了局部极值解的吸引域对遗传算法收敛结果的影响,以及参数搜索空间规模对遗传算法搜索性能的影响,所得结论充分说明了对解空间进行合理划分的必要性。结合正交设计法,提出了基于解空间合理划分的遗传算法,通过仿真实验验证了该算法对多模态优化问题的有效性。
4.分析了两类基于遗传算法无限种群模型的复杂进化系统,在此基础上构造了两类新的种群进化系统,并推导出了它们在单基因位情况下的动力方程。对新系统在稳态情况下的复杂动力行为进行了分析,通过相图、分岔图和Lyapunov指数谱图,严格地证明了其中一类系统具有混沌特性,而另一类系统具有复杂的周期特性。根据以上结论,提出了一种能够持续保持遗传算法的种群多样性的方法,它对求解多模态优化问题很有帮助。
How to apply genetic algorithms to solve multi-modal optimization problems has been a focus in the research community of genetic algorithm and evolutionary computation. It is also fundamental to the basic theory and industrial applications of genetic algorithms. There have been many successful cases in real world, but the existing theoretic analyses are far from perfect. The influence of the external parameters on the efficiency of genetic algorithms is especially investigated, and the theoretic analyses are made and extended corollaries are derived. The theoretical achievements are used to the design of high-efficiency genetic algorithms for solving multi-modal optimization problems. The main contents of the dissertation are as follows:
1. The development of the multi-modal optimization and genetic algorithms is summarized, and the current literatures of the application of genetic algorithms to the multi-modal optimization problems are reviewed. The basic theories of genetic algorithms are introduced, and the development trends of genetic algorithms are evaluated.
2. The infinite population model of the gene pool genetic algorithm (GPGA) is built based on the Walsh transform, and the relation between the fitness difference of local optima of the BINEEDLE fitness function and the efficiency of GPGA is defined. The application scope of the theoretical corollaries are analyzed and proved via simulation tests. According to the theoretical propositions, we invent the two-step genetic algorithm for the multi-modal optimization problems, and get satisfactory empirical results. It also proves the theoretic corollaries indirectly.
3. Both the influence of the attraction basins of the local optima on the convergence of genetic algorithms and the influence of the solution space spectrum on the efficiency of GPGA are investigated, which illustrates the significance of partition of the solution space. According to the orthogonal design, we propose a new genetic algorithm based on the rational partition of the solution space. The simulation experiments show that the improved algorithm converges to the global optimum more quickly, and can find more local optima than traditional methods on multi-modal optimization problems.
4. Two complex evolution systems are analyzed based on infinite population model of the genetic algorithm. According to the analytical results, two new evolution systems are constructed, and the iterative equations of the dynamical system models of both evolution systems is derived in one-bit case. The complex behaviors of both systems in steady state are investigated by phase portraits, bifurcation diagrams and Lyapunov-exponent spectrums, which show that the first evolution system is a chaotic system and the second evolution system contains complex periodic behaviors. According to these conclusions, we present the method to keep the diversity of the population of the genetic algorithm, which is good for finding more useful solutions of multi-modal optimization problems.
1.对多模态优化问题和遗传算法进行了简单回顾,对应用遗传算法解决多模态优化问题的研究现状进行了分析,介绍了遗传算法的基础理论及其发展趋势。
2.对经Walsh变换后的基因池遗传算法的无限种群模型进行了分析,刻画了“双峰函数”局部极值解的适值差与基因池遗传算法的搜索性能之间的关系,对上述理论结果进行了理论推广和实验验证。最后,根据所得结论设计出了针对多模态优化问题的两阶段遗传算法,并得到了满意的实验结果,这也从侧面说明了上述理论结果的正确性。
3.讨论了局部极值解的吸引域对遗传算法收敛结果的影响,以及参数搜索空间规模对遗传算法搜索性能的影响,所得结论充分说明了对解空间进行合理划分的必要性。结合正交设计法,提出了基于解空间合理划分的遗传算法,通过仿真实验验证了该算法对多模态优化问题的有效性。
4.分析了两类基于遗传算法无限种群模型的复杂进化系统,在此基础上构造了两类新的种群进化系统,并推导出了它们在单基因位情况下的动力方程。对新系统在稳态情况下的复杂动力行为进行了分析,通过相图、分岔图和Lyapunov指数谱图,严格地证明了其中一类系统具有混沌特性,而另一类系统具有复杂的周期特性。根据以上结论,提出了一种能够持续保持遗传算法的种群多样性的方法,它对求解多模态优化问题很有帮助。
How to apply genetic algorithms to solve multi-modal optimization problems has been a focus in the research community of genetic algorithm and evolutionary computation. It is also fundamental to the basic theory and industrial applications of genetic algorithms. There have been many successful cases in real world, but the existing theoretic analyses are far from perfect. The influence of the external parameters on the efficiency of genetic algorithms is especially investigated, and the theoretic analyses are made and extended corollaries are derived. The theoretical achievements are used to the design of high-efficiency genetic algorithms for solving multi-modal optimization problems. The main contents of the dissertation are as follows:
1. The development of the multi-modal optimization and genetic algorithms is summarized, and the current literatures of the application of genetic algorithms to the multi-modal optimization problems are reviewed. The basic theories of genetic algorithms are introduced, and the development trends of genetic algorithms are evaluated.
2. The infinite population model of the gene pool genetic algorithm (GPGA) is built based on the Walsh transform, and the relation between the fitness difference of local optima of the BINEEDLE fitness function and the efficiency of GPGA is defined. The application scope of the theoretical corollaries are analyzed and proved via simulation tests. According to the theoretical propositions, we invent the two-step genetic algorithm for the multi-modal optimization problems, and get satisfactory empirical results. It also proves the theoretic corollaries indirectly.
3. Both the influence of the attraction basins of the local optima on the convergence of genetic algorithms and the influence of the solution space spectrum on the efficiency of GPGA are investigated, which illustrates the significance of partition of the solution space. According to the orthogonal design, we propose a new genetic algorithm based on the rational partition of the solution space. The simulation experiments show that the improved algorithm converges to the global optimum more quickly, and can find more local optima than traditional methods on multi-modal optimization problems.
4. Two complex evolution systems are analyzed based on infinite population model of the genetic algorithm. According to the analytical results, two new evolution systems are constructed, and the iterative equations of the dynamical system models of both evolution systems is derived in one-bit case. The complex behaviors of both systems in steady state are investigated by phase portraits, bifurcation diagrams and Lyapunov-exponent spectrums, which show that the first evolution system is a chaotic system and the second evolution system contains complex periodic behaviors. According to these conclusions, we present the method to keep the diversity of the population of the genetic algorithm, which is good for finding more useful solutions of multi-modal optimization problems.
引文
[1]李敏强,寇纪淞,林丹,李书全著.遗传算法的基本理论与应用.北京:科学出版社,2002年.
[2]顾培亮编著.系统分析与协调.天津:天津大学出版社,1998.
[3]郝翔,李人厚.一种快速有效的多模态函数寻优方法.控制理论与应用,1997,vol.14(5),765-769.
[4]刘洪杰,王秀峰,王治宝.改进的多模态遗传算法及其在投资组合中的应用.控制与决策,2003,vol.18(2),173-176.
[5]杨慧珠,张世俊,杜祥.小生境遗传算法求解多峰问题在反演中应用.地球物理学进展,2001,Vol.16(2),35-41.
[6]贾红伟.区域性两阶段演化算法在多峰函数优化中的应用.集美大学学报,2005,Vol.10(3),227-230.
[7]徐东亮.基于优育子群迁徙策略的多模态遗传算法研究.集美大学学报,2005,Vol.35(5),88-92.
[8]杨孔雨,王秀峰.免疫记忆遗传算法及其完全收敛性研究.计算机工程与应用,2005,Vol.41(12),47-50.
[9]Himmelblau D M著,张义炎等译.北京:科学出版社,1981.
[10]周济编.机械设计优化方法及应用.北京:高等教育出版社,1989.
[11]吕佳,熊忠阳.面向多模态函数优化的混沌免疫网络算法研究.计算机应用,2006,Vol.26(2),456-459.
[12]王小平,曹立明著.遗传算法—理论、应用与软件实现.西安:西安交通大学出版社,2002.
[13]王晓兰,李恒杰.多模态函数优化的小生境克隆选择算法.甘肃科学学报,2006,Vol.18(3),64-69.
[14]刘勇,康力山,陈毓屏著.非线性并行算法-遗传算法.北京:科学出版社,1998.
[15]刘勇,康力山,陈毓屏著.非线性并行算法-模拟退火算法.北京:科学出版社,1998.
[16]郑高飞,王秀峰.带子群自组织蠕虫算法及其在多模态问题中的应用.计算机工程,2006,Vol.32(7),182-184.
[17]葛红.免疫算法综述.华南师范大学学报,2002,Vol.8(3),120-126.
[18]黄挚雄,张登科,黎群辉.蚁群算法及其改进形式综述.计算技术与自动化,2006,Vol.25(3),35-38.
[19]谢晓峰,张文俊,杨之廉.微粒群算法综述.控制与决策,2003,Vol.18(2),129-134.
[20]康琦,张燕,汪镭,吴启迪.群体智能应用综述.冶金自动化,2005,Vol.5(1),7-10.
[21]Angeline P J.Evolutionary optimization versus particle swarm optimization:Philosophy and performance difference.Proc of the 7th Annual Conf,On Evolutionary Programming,Washington DC.Germany:Springer,1998.601-610
[22]Shi Yuhui,Eberhart R.Parameter selection in particle swarm optimization.Proc of the 7th Annual Conf,On Evolutionary Programming,Washington DC.Germany:Springer,1998.591-600
[23]杨海军,进化计算中的模式理论、涌现及应用研究,天津大学博士学位论文,2003年10月.
[24]Holland J H.Outline for a logical theory of adaptive systems.Journal of the Association for Computing Machinery,1962,9(3):297-314.
[25]Goldberg D E.Genetic Algorithms in search,optimization and machine learning.MA:Addison-Wesley Publishing Company,1989.
[26]De Jong K A.An analysis of the behavior of a class of genetic adaptive systems(Ph.D Dissertation).University of Michigan,No.76-9381,1975.
[27]Bagley J D.The behavior of adaptive systems which employ genetic and correlation algorithms(Ph.D Dissertation).University of Michigan,No.68-7556,1967.
[28]Rosenberg R S.A computer simulation of a biological population,Unpublished Manuscript,1966.
[29]Cavicchio Jr D J.Adaptive search using simulation evolution(Ph.D Dissertation).University of Michigan,No.71-0199,1970.
[30]Weinberg R.Computer simulation of a living cell(Ph.D Dissertation).University &Michigan,No.71-4766,1970.
[31]Holland J H.Adaptation plans optimal for payoff-only environments.Proc of the 2th Hawaii Int Conf,On System Sciences,Hawaii.Germany:Springer,917-920,1969.
[32]Holland J H.Adaptation in natural and artificial systems.Cambridge,Massachusetts:MIT Press,1992.
[33]Goldberg D E.Computer-aided gas pipeline operation using genetic algorithm and rule learning.Doctoral dissertation,No.8402282,University of Michigan,1983.
[34]Goldberg D E.Genetic Algorithms in Search,Optimization,and Machine Learning.MA,1989.
[35]Davis L.Handbook of genetic algorithms.New York:Van Nostrand Reinhold,1991.
[36]De Jong K A.Adapative system design:a genetic approach.IEEE Transaction Systems,Man,Cybern.1998,SMC10,566-574.
[37]Fonseca C M,Fleming P J.Genetic Algorithms for Multi-objective optimization:formulation,discussion and generalization.Pro.5th Int.Conf,On Genetic Algorithms, 1993,416-423.
[38] Kim J-H, Shim H-S. Evolutionary programming-based optimal robust locomotion control of autonomous mobile robots. Proc. 4th Ann. Conf, On Evolutionary Programming, 1995, 631-644.
[39] Keymeulen D, Masaya I, Yasuo K et al. Online Evolution for a Self-Adapting Robotic Navigation System Using Evolvable Hardware. Artificial Life 1998, 4(4), 359-393.
[40] Zhang J-X, Tang W-S, Xu Y. Robust H_∞ Output Feedback Controller Design for Uncertain Piecewise Linear Systems Based on Genetic Algorithm. Proc. First BIC-TA, Wu Han, 2006, 457-458.
[41] Goldberg D E, Lingle R. Alleles, loci and the traveling salesman problem. In Proceedings of the Second International Conference on Genetic Algorithms, Mahwah, NJ. Lawrence Eribaum Associate, 1985.
[42] Greffenstette J, Gopal R, Rosmaita B et al. Genetic Algorithms for the Travelling Salesman Problem. Proceedings of the First International Conference on Genetic Algorithms, Lawrence Eribaum, 1985, 160-168.
[43] Fogel D B. An evolutionary approach to the traveling salesman problem. Biol.Cybernet. 1988, 60, 139-144.
[44] Fogel D B. A Parallel Processing Approach to a Multiple Traveling Salesman Problem Using Evolutionary Programming. Proc. of the Fourth Annual Symposium on Parallel Processing, Fullerton, CA, 1990, 318-326.
[45] Fogel D B. Applying Evolutionary Programming to Selected Traveling Salesman Problems. Cybernetics and Systems, 1993, Vol.24, 27-36.
[46] Davis L. Job shop scheduling with genetic algorithms. Proceedings of the First International Conference on Genetic Algorithms. Lawrence Eribaum, Hillsdale, 1985. 136-140
[47] Yamada T, Nakano R. A Genetic Algorithm Applicable to Large-Scale Job-Shop Problems. Proc. of the Second International Conference on Parallel Problem Solving from Nature PPSN '92, 1992. 281-290
[48] Fogel D B, Fogel L J. Using evolutionary programming to schedule tasks on a suite of heterogeneous computers, Comput. Operat Res. 1996, 527-534.
[49] Thangia S R, Vinayagamoorthy R, Gubbi A V. Vehicle routing with time deadlines using genetic and local algorithms, Oroc. 5th Int. Conf. On Genetic Algorithms, San Mateo, CA: Morgan Kaufmann, 1993, 506-513.
[50] Machado P, Tavares J, Pereira F B, et al. Vehicle Routing Problem: Doing it the Evolutionary Way. In Proceedings of the Genetic and Evolutionary Computation Conference, New York, 2002. 690-699
[51] Bauer R J. Genetic algorithms and Investment Strategies. New York: Wiley, 1994.
[52] Allen F, Karjalainen R. Using genetic algorithms to find technical trading rules. Journal of Financial Economics,1999,51,245-271.
[53]赵衍刚,江近仁.一个以遗传算法为基础的结构可靠性分析方法.地震工程与工程振动.1995,15(3):47-56.
[54]Jenkins W M.Structural Optimization with the Genetic Algorithm.Structural Engineer,1991,69(24).408-422.
[55]Li S-Q,Kou J-S,Li M-Q.Real Estate Investment Decision Method Based on Genetic Algorithm.4th International Conference Industrial Engineering and Engineering Management,Hong Kong,1997.
[56]李书全,李敏强,寇纪淞.遗传规划及其在地基设计中的应用.系统工程学报.1997(7),65-82.
[57]Huang G H,et al.Development of Grey Critical Path Method for Construction.Engineering Optimization,28(28),157-174.
[58]Dragan A S,et al.Genetic Algorithms for Least-Cost Design of Water Distribution Networks.Journal of Water Resources Planning and Management,1997,123(2),67-77.
[59]王黎,马光文.基于遗传算法的水电站优化调度新方法.系统工程理论与实践,1997(7),65-82.
[60]Grimes C A.Application of Genetic Algorithms for Least-Cost Design of Water Distribution Networks.Journal of Water Resources Planning and Management,1997,123(2),67-77.
[61]Leehter Yao,et al.Nonlinear Parameter Estimation via the Genetic Algorithm.IEEE Transactions on Signal Processing,1994(80),213-234.
[62]王熙法等.遗传算法在模式识别中的应用.小型计算机系统,1997,18(10),32-36.
[63]Thomas Back,Fogel D B and Zbigniew Michalewicz.Evolutionary Computation 1 Basic Algorithms and Operators.Bristol and Philadelphia:Institute of Physics Publishing,2000.
[64]Schwefel H P.Kybernetische Evolution als Strategie der experimentllen Forschung in der Str0mungstechnik Dipl.-Ing.Thesis,Technical University Berlin,1965.
[65]Bienert P.aufbsu einer Optimierungsautomatik fUr drei parameter Dipl.-Ing.Thesis,Technical University Berlin,1967.
[66]Rechenberg I.Evolutionsstrategie:Optimierung trchnischer Systeme nach Prinzipien der biologischen Evolution.Stuttgart:Frommann-Holzboog,1973.
[67]Schwefel H P.Numerical Optimization of Computer Models.Chichester:Wiley,1981.
[68]Beyer H G The Theory of Evolution Strategies.Berlin Heidelberg:Springer-Verlag,2001.
[69]Beyer H G.Toward a theory of evolution strategies:On the benefit of sex-the (μ/μ,λ) -theory.Evolutionary Computation,1995,3,81-11.
[70]Fogel L J,Owens A J,Walsh M J.Artificial Intelligence through Simulated Evolution,New York:Wiley,1966.
[71]Fogel D B.An evolutionary programming for voice feature analysis Proc.23rd Asilomar Conference on Signals.System and Computers,1989,26-33.
[72]Fogel D B.Evolutionary Computation:Toward a New Philosophy of Machine Intelligence.New York:IEEE,1995.
[73]Fogel D B.Evolutionary computation:the fossil record.Piscataway,NJ:IEEE Press,1998.
[74]Koza J R.Genetic Programming 1.Cambridge,MA:The MIT Press,1992.
[75]Koza J R.Genetic Programming 2.Cambridge,MA:The MIT Press,1994.
[76]王雪梅,王义和.模拟退火算法与遗传算法的结合.计算机学报,Vol.20(4),1997.381-384
[77]王喆,顾培亮,梁皎洁,钱卫国.模拟退火改进遗传算法在股市投资决策中的应用.天津大学学报,Vol.33(1),2000.81-84
[78]Thomas Back.An Overview of Parameter Control Methods by Self-Adaptation in Evolutionary Algorithms.Fundamenta Informaticae archive,Vol.35,1998.51-66
[79]刘虎,张焕春,经亚枝,朱力立.并行多种群模糊遗传算法参数.吉林大学学报,Vol.23(6),2005.629-634.
[80]De Jong K A.Genetic Algorithms:a 25 year perspective.In Proceedings of the fifth International Conference on Genetic Algorithms,Whitley L D(ed.),San Mateo,CA:Morgan Kaufmann,1993:169-177.
[81]Mengshoel O L,Goldberg D E.Probabilistic crowding:deterministic crowding with probabilistic replacement.In Proceedings of the Genetic and Evolutionary Computation Conference 1999,Banzhaf W,et al(eds.),San Fransisco,CA:Morgan Kaufmann,1999.
[82]Goldberg D E,Richardson J.Genetic algorithms with sharing for multi-modal function optimization.In Proceedings of the Second International Conference on Genetic Algorithm,Grefenstette J J(ed.),Hillsdale.NJ,Lawrence Erlbaum Associates,1987,41-49.
[83]Beasly D,Bull D R,Martin R R.A sequential niche technique for multi-modal function optimization.Evolutionary Computation,1993,1(2),101-125.
[84]Harik G.Finding multiple solution in problem of bounded difficulty.IlliGal report no.94002,University of Illinois,The Illinois Genetic algorithms Laboratory,Urbana-Champaign,1994.
[85]李敏强,寇纪淞.多模态函数优化的协同多种群遗传算法.自动化学报,Vol.28(4),2002.497-504
[86]张文修,梁怡编著.遗传算法的数学基础.西安,西安交通大学出版社,2000年.
[87]Radcliffe N J.The algebra of genetic algorithms.Annals of Maths and artificial Intelligence,1994,10(1),339-384.
[88] Vose M D, Wright A H. Form Invariance and Implicit Parallelism. Evolutionary Computation, 1999, 9(3). 355-370.
[89] Nix A E, Vose M D. Modeling genetic algorithms with Markov chains. Annals of Mathematics and Artificial intelligence, 1992, 5: 79-88.
[90] Eiben A E, Schippers C A. On Evolutionary Exploration and Exploitation. In A.E.Eiben and Z.Michalewicz editors, Evolutionary Computation, IOS press, 1999, 35-51.
[91] Deaton R, Garzon M, Rose J, Francechetti D R and Stevens S E Jr. DNA Computing : A Review. Evolutionary Computation, IOS press, 1999, 231-247.
[92] Holland J H. Building Blocks, Cohort Genetic Algorithms, and Hyperplane-Defined Functions, Evolutionary Computation, Massachusetts Institute of Technology, 2000, 373-391.
[93] Altenberg L.The Schema Theorem and Price's Theorem. In Whitley,D and Vose,M.D editors, Foundations of Genetic algorithms 3, 1995, 23-49.
[94] Stephens C R, Waelbroeck H. Effective degree of freedom in genetic algorithms and the block hypothesis. In Thomas Back, editor, Proceedings of the Seventh International Conference on Genetic Algorithms, San Mateo: Morgan Kaufmann, 1997, 34-40.
[95] Stephens C R, Waelbroeck H. Analysis of the Effective Degrees of Freedom in Genetic Algorithms Phys. Rev. 1998, E57, 3251-3264.
[96] Stephens C R, Waelbroeck H. Schemata Evolution and Building Blocks. Evolutionary Computation 7(2), 1999, 109-124.
[97] Stephens C R, Waelbroeck H. Schemata as Building Blocks: Does Size Matter? In Wolfgang Banzhaf and Colin Reeves, Foundations of Genetic algorithms 5, San Francisco: Morgan Kaufmann, 1999, 117-135.
[98] Poli R. Exact Schema Theory for Genetic Programming and Variable-Length Genetic Algorithms with One-Point Crossover. Genetic Programming and Evolvable Machines, Kluwer Academic Publishers. Manufactured in The Netherlands, 2001, 2, 123-163.
[99] Poli R, Langdon W B, Reilly U-M O'. Analysis of schema variance and short term extinction likelihoods. In Genetic programming 1998: Proc.Third Annually Conference, University of Wisconsin, Madison, Wisconsin, USA, San Francisco: Morgan Kaufmann, 1998, 284-292.
[100] Poli R. Hyperschema Theory for GP with One-Point Crossover, Building Blocks, and Some New Results in GA Theory. EuroGP2000: 2000, 163-180.
[101] 杨海军,李敏强．模式的形式不变性及准确的模式理论．中国科学(E辑),Vol．33(8),2003．707—714
[102] Bethke AD. Genetic Algorithms as Function Optimizers. Dissertation Abstracts International, 1981, 41(9), 3503(B).
[103] Goldberg D E. Construction of high-order deceptive functions using low-order Walsh coeffcients. Annals of Mathematics and Artificial Intelligence, 1992(5). 35-48.
[104] Deb K, Goldberg D E. Sufficient Conditions for Deceptive and Easy Binary Functions. Evolutionary Computation, 2(3), 1992. 279-311
[105] Pelikan M, Goldberg D E. Escaping Hierarchical Traps with Competent Genetic Algorithms. IlliGAL Report No. 2001003, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2001.
[106] Sastry K, Pelikan M, Goldberg D E. Combining Competent Crossover and Mutation Operators: A Probabilistic Model Building Approach. IlliGAL Report No. 2004010, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2004.
[107] Lima C F, Sastry K, Goldberg D ,E Lobo F G. Combining Competent Crossover and Mutation Operators: A Probabilistic Model Building Approach. IlliGAL Report No. 2005002, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2005.
[108] 李敏强,寇纪淞．遗传算法的模式欺骗性分析．中国科学(E辑),Vol．32(1),2002． 95—102
[109] Whitley L D. Fundamental principles of deception in genetic search, In Foundations of Genetic algorithms, Rawlins G(ed.), San Mateo, CA: Morgan Kaufmann, 1991:221-241.
[110] Homaifar A, Qi X, Fost J. Analysis and design of a general GA deceptive problem. In Proceedings of the Fourth International Conference on Genetic Algorithms(ICGA4), Belew R, Brooker L.(eds.), San Francisco: Morgan Kaufmann Publishers, Inc., 1991. 196-203
[111] Wright A H, Jonathan E R, James R N. Analysis of the Simple Genetic Algorithm on the Single-peak and Double-peak Landscapes Proceedings of CEC 2002, IEEE Press.
[112] Harik G. Linkage Learning via Probabilistic Modeling in the ECGA. IlliGAL Report No. 99010, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 1999.
[113] Chen Y-P, Goldberg D E. Convergence Time for the Linkage Learning Genetic Algorithm. IlliGAL Report No. 2003025, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2003.
[114] Sastry K, Pelikan M, Goldberg D E. Efficiency Enhancement of Genetic Algorithms via Building-Block-Wise Fitness Estimation. IlliGAL Report No. 20040010, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2004.
[115] Yu T L, Goldberg D E. Quality and Efficiency of Model Building for Genetic Algorithms. IlliGAL Report No. 2004004, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2004.
[116] Chen Ying-ping. Extending the Scalability of Linkage Learning Genetic Algorithms: Theory and Practice. IlliGAL Report No. 2004018, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2004.
[117] Sastry K, Pelikan M, Goldberg D E. Efficiency Enhancement of Probabilistic Model Building Genetic Algorithms. IlliGAL Report No. 20040020, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2004.
[118] Butz M V, Pelikan M, Xavier Llora, Goldberg D E. Extracted Global Structure Makes Local Building Block Processing Effiective in XCS. IlliGAL Report No. 20050011, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2005.
[119] Butz M V, Pelikan M, Xavier Llora, Goldberg D E. Automated Global Structure Extraction For Effiective Local Building Block Processing in XCS. IlliGAL Report No. 20050010, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2005.
[120] Yu T-L, Sastry K, Goldberg D E. Linkage Learning, Overlapping Building Blocks, and Systemmatic Strategy for Scalable Recombination. IlliGAL Report No. 20050016, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2005.
[121] Deb K, Goldberg D E. MGA in C: A Messy Genetic Algorithm in C. IlliGAL Report No. 91008, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 1991.
[122] Munetomo M, Goldberg D E. Identifying Linkage by Nonlinearity Check. IlliGAL Report No. 98012, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 1998.
[123] Munetomo M, Goldberg D E. Designing a genetic algorithm using the identifying linkage by nonlinearity check. IlliGAL Report No. 98014, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 1998.
[124] Schaffer J D, Morishima A. An adaptive crossover distribution mechanism for genetic algorithms. Proceedings of the Second International Conference on Genetic Algorithms(ICGA-87), 1987. 36-40
[125] Levenick J R. Metabits: Genetic endogenous crossover control. Proceedings of the sixth International Conference on Genetic Algorithms(ICGA-95), 1995. 88-95
[126] Smith J, Fogarty T C. An adaptive poly-parental recombination strategy. Proceedings of AISB-95 Workshop on Evolutionary computing, 1995. 48-61
[127] Smith J, Fogarty T C. Recombination strategy adaptation via evolution of gene linkage. Proceedings of the 1996 IEEE International Conference on Evolutionary Computation, 1996. 826-831
[128] Baluja S. Population-based incremental learning: A method for integrating genetic search based function optimization and competitive learning. Tech. Rep. No. CMU-CS-94-163. Pittsburg, PA: Carnegie Mellon University.
[129] Harik G R, Lobo F G, Goldberg D E. The compact genetic algorithm. Proceedings of the 1998 IEEE International Conference on Evolutionary Computation,1998.523-528
[130]Harik G R,Lobo F G,Goldberg D E.The compact genetic algorithm.IEEE Transactions on Evolutionary Computation,3(4),287-297.
[131]Pelikan M,Goldberg D E,Cantu-Paz E.Linkage problem,distribution estimation,and Bayesian networks.Evolutionary Computation,8(3),311-341.
[132]Yu T-L,Goldberg D E,Yassine A,Chen Y-P.A Genetic Algorithm Design Inspired by Organizational Theory:A Pilot Study of a Dependency Structure Matrix Driven Genetic Algorithm.IlliGAL Report No.2003007,University of Illinois at Urbana-Champaign,Illinois Genetic Algorithms Laboratory,Urbana,2003.
[133]Yu T-L.A Matrix Approach for Finding Extrema:Problems with Modularity Hierarchy and Overlap.IlliGAL Report No.2007012,University of Illinois at Urbana-Champaign,Illinois Genetic Algorithms Laboratory,Urbana,2007.
[134]钟开莱著,刘文译.概率论教程.上海:上海科学技术出版社,1989年.
[135]施仁杰.马尔科夫链基础及其应用.西安,西安电子科技大学出版社,1992.
[136]Rudolph G.Finite Markov Chain Results in Evolutionary Computation:A Tour d'Horizon.Fundamenta Informaticae archive,Vol.35,1998.67-89
[137]Vose M D.The simple genetic algorithms:Foundations and theory.Cambridge,Massachusetts:The MIT Press,1999.
[138]Jong K A D,Spears W M,Gordon D F.Using markov chains to analyze GAFOs.Foundations of genetic algorithms 3,Whitley L D,Vose M D(eds.),San Mateo.CA:Morgan Kaufmann,1995.
[139]Rudolph G.Convergence rates of evolutionary algorithms for a class of convex objective functions,Control and Cybernetics,1997,26(3),375-390.
[140]Mahfoud S W,Goldberg D E.Parallel recombinative simulated annealing:A genetic algorithm.Parallel Computing,1995,21(1),1-28.
[141]张讲社,徐宗本,梁怡.整体退火遗传算法及其收敛充要条件.中国科学(E辑),1997,Vol.27(2),154-164.
[142]Cerf R.The dynamics of mutation-selection algorithms with large population sizes.Annales de l'Institut Henri Poincare,Probabilites et Statistiques,32(4),1996.778-817
[143]Suzuki J.A further result on the markov chain model of genetic algorithm and its application to a simulated annealing-like strategy.IEEE Transactions on Systems,Man,and Cybernetics-Part B,28(1),1998.95-102
[144]Lozano J A,Larranaga P,Grana M,Albizuri F X.Genetic algorithms:Bridging the Convergence gap.Theoretical Computer Science,229(1),1999.11-22
[145]He J,Kang K.On the convergence rates of genetic algorithms.Theoretical Computer Science,229(1),1999.23-34
[146]Vose M D,Liepins G E.Punctuated equilibria in Genetic search.Complex Systems,1991,Vol(5),3-144.
[147]Liepins G E,Vose M D.Characterizing Crossover in Genetic Algorithms.Annals of Mathematics and Artificial Intelligence,1992,5(1),27-34.
[148]Vose M D,Wright A H.Simple Genetic Algorithms with Linear Fitness,Evolutionary Computation,1994,2(4),347-368.
[149]Vose M D.Modeling simple genetic algorithms.Evolutionary Computation,1996,3,453-472.
[150]Vose M D.Random Heuristic Search,Theor.Comput.Sci.1999,229(1),103-142.
[151]Rowe J E,Vose M D,Wright A H.Group Properties of Crossover and Mutation.Evolutionary Computation,2002,10,151-184.
[152]Wright A H,Rowe J E,Poli R,C R Stephens.A Fixed Point Analysis of a Gene Pool GA with Mutation.The Genetic and Evolutionary Computation Conference (GECCO-2002),Langdon W B,et al(eds.).San Francisco:Morgan Kaufmann:2002.642-649
[153]Wright A H.Bistability of the Needle Function in the Presence of Truncation Selection.Springer-Verlag Lecture Notes in Computer Science,2004,LNCS3103,330-342.
[154]Shukla P K,Deb K.On finding multiple Pareto-optimal solutions using classical and evolutionary generating methods.European Journal of Operational Research,2007,181(3),1630-1652.
[155]Wright A H et al.Bistability in a Gene Pool GA with Mutation.In:Kenneth A,De Jong(Ed.) Foundations of Genetic Algorithms 7,Terremolinos Spain.CA:Morgan Kaufmann,2003.63-80
[156]Vose M D,Wright A H.The Simple Genetic Algorithm and the Walsh Transform:Part Ⅰ,Theory,Evolutionary Computation,1998,6(3),253-273.
[157]Vose M D,Wright A H.The Simple Genetic Algorithm and the Walsh Transform:Part Ⅱ,The Inverse,Evolutionary Computation,1998,6(3),275-289.
[158]姚妙新,陈芳启.非线性理论数学基础.天津:天津大学出版社,2005,71-92.
[159]Geiringer H.On the Probability of Linkage in Mendelian Heredity.Annals of Mathematical Statistics 15,1944.25-57
[160]IWAMATSU M.Multi-specie particle swarm optimizer for multimodal function optimization.IEICE Transactions on Information and Systems,2006,E89-D(3),1181-1187.
[161]姬振豫.正交设计的方法与理论.香港:世界科技出版社,2001.
[162]郝翔,李人厚.一种快速有效的多模态函数寻优方法—双群体遗传算法.控制理论与应用,1997,10(5),765-768.
[163]Hastings A.Stable cycling in discrete-time genetic models.Proc.Nat.Acad.Sci.USA1981.7224-7225
[164]Wright A H,Agapie A.Cyclic and Chaotic Behavior in Genetic Algorithms.The Genetic and Evolutionary Computation Conference GECCO-2001,Goodman E K,et al(eds.).Morgan Kaufmann:2001.718-724
[165]Liepins G E,Vose M D.Deceptiveness and genetic algorithms dynamics.In Rawlins G J E(Ed.) Foundations Of Genetic Algorithms 1,San Mateo,CA:Morgan Kaufmann Publishers,Inc.1991.36-50
[166]Beyer,H.G On the Dynamics of EAs without Selection.Foundations of Genetic Algorithms 5,Wolfgang Banzhaf and Colin Reeves,San Francisco:Morgan Kaufmann Press,1999.5-26
[167]Juliany J,Vose M D.The genetic algorithm fractal.Evolutionary Computation,1994,2(2),165-180.
[168]Collard P,Clergue M.Genetic Algorithm:from Hegemony to Chaos.Complex Systems,12(1),2000.1-29
[169]龙运佳.混沌工程学.中国工程科学,2001年,Vol.3(2),1-7.
[170]陈士华,陆君安.混沌动力学.武汉:武汉水利电力大学出版社,1999.
[171]王杰智,陈增强,袁著祉.一类新的混沌系统及其性质研究.物理学报,Vol.55(8),2006.3956-3963
[172]Wang Jie-Zhi,Chen Zeng-Qiang,Yuan Zhu-Zhi.The generation of a hyperchaotic system based on three-dimensional autonomous chaotic system.Chinese Physics,Vol.15(6),2006.1216-1225
[2]顾培亮编著.系统分析与协调.天津:天津大学出版社,1998.
[3]郝翔,李人厚.一种快速有效的多模态函数寻优方法.控制理论与应用,1997,vol.14(5),765-769.
[4]刘洪杰,王秀峰,王治宝.改进的多模态遗传算法及其在投资组合中的应用.控制与决策,2003,vol.18(2),173-176.
[5]杨慧珠,张世俊,杜祥.小生境遗传算法求解多峰问题在反演中应用.地球物理学进展,2001,Vol.16(2),35-41.
[6]贾红伟.区域性两阶段演化算法在多峰函数优化中的应用.集美大学学报,2005,Vol.10(3),227-230.
[7]徐东亮.基于优育子群迁徙策略的多模态遗传算法研究.集美大学学报,2005,Vol.35(5),88-92.
[8]杨孔雨,王秀峰.免疫记忆遗传算法及其完全收敛性研究.计算机工程与应用,2005,Vol.41(12),47-50.
[9]Himmelblau D M著,张义炎等译.北京:科学出版社,1981.
[10]周济编.机械设计优化方法及应用.北京:高等教育出版社,1989.
[11]吕佳,熊忠阳.面向多模态函数优化的混沌免疫网络算法研究.计算机应用,2006,Vol.26(2),456-459.
[12]王小平,曹立明著.遗传算法—理论、应用与软件实现.西安:西安交通大学出版社,2002.
[13]王晓兰,李恒杰.多模态函数优化的小生境克隆选择算法.甘肃科学学报,2006,Vol.18(3),64-69.
[14]刘勇,康力山,陈毓屏著.非线性并行算法-遗传算法.北京:科学出版社,1998.
[15]刘勇,康力山,陈毓屏著.非线性并行算法-模拟退火算法.北京:科学出版社,1998.
[16]郑高飞,王秀峰.带子群自组织蠕虫算法及其在多模态问题中的应用.计算机工程,2006,Vol.32(7),182-184.
[17]葛红.免疫算法综述.华南师范大学学报,2002,Vol.8(3),120-126.
[18]黄挚雄,张登科,黎群辉.蚁群算法及其改进形式综述.计算技术与自动化,2006,Vol.25(3),35-38.
[19]谢晓峰,张文俊,杨之廉.微粒群算法综述.控制与决策,2003,Vol.18(2),129-134.
[20]康琦,张燕,汪镭,吴启迪.群体智能应用综述.冶金自动化,2005,Vol.5(1),7-10.
[21]Angeline P J.Evolutionary optimization versus particle swarm optimization:Philosophy and performance difference.Proc of the 7th Annual Conf,On Evolutionary Programming,Washington DC.Germany:Springer,1998.601-610
[22]Shi Yuhui,Eberhart R.Parameter selection in particle swarm optimization.Proc of the 7th Annual Conf,On Evolutionary Programming,Washington DC.Germany:Springer,1998.591-600
[23]杨海军,进化计算中的模式理论、涌现及应用研究,天津大学博士学位论文,2003年10月.
[24]Holland J H.Outline for a logical theory of adaptive systems.Journal of the Association for Computing Machinery,1962,9(3):297-314.
[25]Goldberg D E.Genetic Algorithms in search,optimization and machine learning.MA:Addison-Wesley Publishing Company,1989.
[26]De Jong K A.An analysis of the behavior of a class of genetic adaptive systems(Ph.D Dissertation).University of Michigan,No.76-9381,1975.
[27]Bagley J D.The behavior of adaptive systems which employ genetic and correlation algorithms(Ph.D Dissertation).University of Michigan,No.68-7556,1967.
[28]Rosenberg R S.A computer simulation of a biological population,Unpublished Manuscript,1966.
[29]Cavicchio Jr D J.Adaptive search using simulation evolution(Ph.D Dissertation).University of Michigan,No.71-0199,1970.
[30]Weinberg R.Computer simulation of a living cell(Ph.D Dissertation).University &Michigan,No.71-4766,1970.
[31]Holland J H.Adaptation plans optimal for payoff-only environments.Proc of the 2th Hawaii Int Conf,On System Sciences,Hawaii.Germany:Springer,917-920,1969.
[32]Holland J H.Adaptation in natural and artificial systems.Cambridge,Massachusetts:MIT Press,1992.
[33]Goldberg D E.Computer-aided gas pipeline operation using genetic algorithm and rule learning.Doctoral dissertation,No.8402282,University of Michigan,1983.
[34]Goldberg D E.Genetic Algorithms in Search,Optimization,and Machine Learning.MA,1989.
[35]Davis L.Handbook of genetic algorithms.New York:Van Nostrand Reinhold,1991.
[36]De Jong K A.Adapative system design:a genetic approach.IEEE Transaction Systems,Man,Cybern.1998,SMC10,566-574.
[37]Fonseca C M,Fleming P J.Genetic Algorithms for Multi-objective optimization:formulation,discussion and generalization.Pro.5th Int.Conf,On Genetic Algorithms, 1993,416-423.
[38] Kim J-H, Shim H-S. Evolutionary programming-based optimal robust locomotion control of autonomous mobile robots. Proc. 4th Ann. Conf, On Evolutionary Programming, 1995, 631-644.
[39] Keymeulen D, Masaya I, Yasuo K et al. Online Evolution for a Self-Adapting Robotic Navigation System Using Evolvable Hardware. Artificial Life 1998, 4(4), 359-393.
[40] Zhang J-X, Tang W-S, Xu Y. Robust H_∞ Output Feedback Controller Design for Uncertain Piecewise Linear Systems Based on Genetic Algorithm. Proc. First BIC-TA, Wu Han, 2006, 457-458.
[41] Goldberg D E, Lingle R. Alleles, loci and the traveling salesman problem. In Proceedings of the Second International Conference on Genetic Algorithms, Mahwah, NJ. Lawrence Eribaum Associate, 1985.
[42] Greffenstette J, Gopal R, Rosmaita B et al. Genetic Algorithms for the Travelling Salesman Problem. Proceedings of the First International Conference on Genetic Algorithms, Lawrence Eribaum, 1985, 160-168.
[43] Fogel D B. An evolutionary approach to the traveling salesman problem. Biol.Cybernet. 1988, 60, 139-144.
[44] Fogel D B. A Parallel Processing Approach to a Multiple Traveling Salesman Problem Using Evolutionary Programming. Proc. of the Fourth Annual Symposium on Parallel Processing, Fullerton, CA, 1990, 318-326.
[45] Fogel D B. Applying Evolutionary Programming to Selected Traveling Salesman Problems. Cybernetics and Systems, 1993, Vol.24, 27-36.
[46] Davis L. Job shop scheduling with genetic algorithms. Proceedings of the First International Conference on Genetic Algorithms. Lawrence Eribaum, Hillsdale, 1985. 136-140
[47] Yamada T, Nakano R. A Genetic Algorithm Applicable to Large-Scale Job-Shop Problems. Proc. of the Second International Conference on Parallel Problem Solving from Nature PPSN '92, 1992. 281-290
[48] Fogel D B, Fogel L J. Using evolutionary programming to schedule tasks on a suite of heterogeneous computers, Comput. Operat Res. 1996, 527-534.
[49] Thangia S R, Vinayagamoorthy R, Gubbi A V. Vehicle routing with time deadlines using genetic and local algorithms, Oroc. 5th Int. Conf. On Genetic Algorithms, San Mateo, CA: Morgan Kaufmann, 1993, 506-513.
[50] Machado P, Tavares J, Pereira F B, et al. Vehicle Routing Problem: Doing it the Evolutionary Way. In Proceedings of the Genetic and Evolutionary Computation Conference, New York, 2002. 690-699
[51] Bauer R J. Genetic algorithms and Investment Strategies. New York: Wiley, 1994.
[52] Allen F, Karjalainen R. Using genetic algorithms to find technical trading rules. Journal of Financial Economics,1999,51,245-271.
[53]赵衍刚,江近仁.一个以遗传算法为基础的结构可靠性分析方法.地震工程与工程振动.1995,15(3):47-56.
[54]Jenkins W M.Structural Optimization with the Genetic Algorithm.Structural Engineer,1991,69(24).408-422.
[55]Li S-Q,Kou J-S,Li M-Q.Real Estate Investment Decision Method Based on Genetic Algorithm.4th International Conference Industrial Engineering and Engineering Management,Hong Kong,1997.
[56]李书全,李敏强,寇纪淞.遗传规划及其在地基设计中的应用.系统工程学报.1997(7),65-82.
[57]Huang G H,et al.Development of Grey Critical Path Method for Construction.Engineering Optimization,28(28),157-174.
[58]Dragan A S,et al.Genetic Algorithms for Least-Cost Design of Water Distribution Networks.Journal of Water Resources Planning and Management,1997,123(2),67-77.
[59]王黎,马光文.基于遗传算法的水电站优化调度新方法.系统工程理论与实践,1997(7),65-82.
[60]Grimes C A.Application of Genetic Algorithms for Least-Cost Design of Water Distribution Networks.Journal of Water Resources Planning and Management,1997,123(2),67-77.
[61]Leehter Yao,et al.Nonlinear Parameter Estimation via the Genetic Algorithm.IEEE Transactions on Signal Processing,1994(80),213-234.
[62]王熙法等.遗传算法在模式识别中的应用.小型计算机系统,1997,18(10),32-36.
[63]Thomas Back,Fogel D B and Zbigniew Michalewicz.Evolutionary Computation 1 Basic Algorithms and Operators.Bristol and Philadelphia:Institute of Physics Publishing,2000.
[64]Schwefel H P.Kybernetische Evolution als Strategie der experimentllen Forschung in der Str0mungstechnik Dipl.-Ing.Thesis,Technical University Berlin,1965.
[65]Bienert P.aufbsu einer Optimierungsautomatik fUr drei parameter Dipl.-Ing.Thesis,Technical University Berlin,1967.
[66]Rechenberg I.Evolutionsstrategie:Optimierung trchnischer Systeme nach Prinzipien der biologischen Evolution.Stuttgart:Frommann-Holzboog,1973.
[67]Schwefel H P.Numerical Optimization of Computer Models.Chichester:Wiley,1981.
[68]Beyer H G The Theory of Evolution Strategies.Berlin Heidelberg:Springer-Verlag,2001.
[69]Beyer H G.Toward a theory of evolution strategies:On the benefit of sex-the (μ/μ,λ) -theory.Evolutionary Computation,1995,3,81-11.
[70]Fogel L J,Owens A J,Walsh M J.Artificial Intelligence through Simulated Evolution,New York:Wiley,1966.
[71]Fogel D B.An evolutionary programming for voice feature analysis Proc.23rd Asilomar Conference on Signals.System and Computers,1989,26-33.
[72]Fogel D B.Evolutionary Computation:Toward a New Philosophy of Machine Intelligence.New York:IEEE,1995.
[73]Fogel D B.Evolutionary computation:the fossil record.Piscataway,NJ:IEEE Press,1998.
[74]Koza J R.Genetic Programming 1.Cambridge,MA:The MIT Press,1992.
[75]Koza J R.Genetic Programming 2.Cambridge,MA:The MIT Press,1994.
[76]王雪梅,王义和.模拟退火算法与遗传算法的结合.计算机学报,Vol.20(4),1997.381-384
[77]王喆,顾培亮,梁皎洁,钱卫国.模拟退火改进遗传算法在股市投资决策中的应用.天津大学学报,Vol.33(1),2000.81-84
[78]Thomas Back.An Overview of Parameter Control Methods by Self-Adaptation in Evolutionary Algorithms.Fundamenta Informaticae archive,Vol.35,1998.51-66
[79]刘虎,张焕春,经亚枝,朱力立.并行多种群模糊遗传算法参数.吉林大学学报,Vol.23(6),2005.629-634.
[80]De Jong K A.Genetic Algorithms:a 25 year perspective.In Proceedings of the fifth International Conference on Genetic Algorithms,Whitley L D(ed.),San Mateo,CA:Morgan Kaufmann,1993:169-177.
[81]Mengshoel O L,Goldberg D E.Probabilistic crowding:deterministic crowding with probabilistic replacement.In Proceedings of the Genetic and Evolutionary Computation Conference 1999,Banzhaf W,et al(eds.),San Fransisco,CA:Morgan Kaufmann,1999.
[82]Goldberg D E,Richardson J.Genetic algorithms with sharing for multi-modal function optimization.In Proceedings of the Second International Conference on Genetic Algorithm,Grefenstette J J(ed.),Hillsdale.NJ,Lawrence Erlbaum Associates,1987,41-49.
[83]Beasly D,Bull D R,Martin R R.A sequential niche technique for multi-modal function optimization.Evolutionary Computation,1993,1(2),101-125.
[84]Harik G.Finding multiple solution in problem of bounded difficulty.IlliGal report no.94002,University of Illinois,The Illinois Genetic algorithms Laboratory,Urbana-Champaign,1994.
[85]李敏强,寇纪淞.多模态函数优化的协同多种群遗传算法.自动化学报,Vol.28(4),2002.497-504
[86]张文修,梁怡编著.遗传算法的数学基础.西安,西安交通大学出版社,2000年.
[87]Radcliffe N J.The algebra of genetic algorithms.Annals of Maths and artificial Intelligence,1994,10(1),339-384.
[88] Vose M D, Wright A H. Form Invariance and Implicit Parallelism. Evolutionary Computation, 1999, 9(3). 355-370.
[89] Nix A E, Vose M D. Modeling genetic algorithms with Markov chains. Annals of Mathematics and Artificial intelligence, 1992, 5: 79-88.
[90] Eiben A E, Schippers C A. On Evolutionary Exploration and Exploitation. In A.E.Eiben and Z.Michalewicz editors, Evolutionary Computation, IOS press, 1999, 35-51.
[91] Deaton R, Garzon M, Rose J, Francechetti D R and Stevens S E Jr. DNA Computing : A Review. Evolutionary Computation, IOS press, 1999, 231-247.
[92] Holland J H. Building Blocks, Cohort Genetic Algorithms, and Hyperplane-Defined Functions, Evolutionary Computation, Massachusetts Institute of Technology, 2000, 373-391.
[93] Altenberg L.The Schema Theorem and Price's Theorem. In Whitley,D and Vose,M.D editors, Foundations of Genetic algorithms 3, 1995, 23-49.
[94] Stephens C R, Waelbroeck H. Effective degree of freedom in genetic algorithms and the block hypothesis. In Thomas Back, editor, Proceedings of the Seventh International Conference on Genetic Algorithms, San Mateo: Morgan Kaufmann, 1997, 34-40.
[95] Stephens C R, Waelbroeck H. Analysis of the Effective Degrees of Freedom in Genetic Algorithms Phys. Rev. 1998, E57, 3251-3264.
[96] Stephens C R, Waelbroeck H. Schemata Evolution and Building Blocks. Evolutionary Computation 7(2), 1999, 109-124.
[97] Stephens C R, Waelbroeck H. Schemata as Building Blocks: Does Size Matter? In Wolfgang Banzhaf and Colin Reeves, Foundations of Genetic algorithms 5, San Francisco: Morgan Kaufmann, 1999, 117-135.
[98] Poli R. Exact Schema Theory for Genetic Programming and Variable-Length Genetic Algorithms with One-Point Crossover. Genetic Programming and Evolvable Machines, Kluwer Academic Publishers. Manufactured in The Netherlands, 2001, 2, 123-163.
[99] Poli R, Langdon W B, Reilly U-M O'. Analysis of schema variance and short term extinction likelihoods. In Genetic programming 1998: Proc.Third Annually Conference, University of Wisconsin, Madison, Wisconsin, USA, San Francisco: Morgan Kaufmann, 1998, 284-292.
[100] Poli R. Hyperschema Theory for GP with One-Point Crossover, Building Blocks, and Some New Results in GA Theory. EuroGP2000: 2000, 163-180.
[101] 杨海军,李敏强．模式的形式不变性及准确的模式理论．中国科学(E辑),Vol．33(8),2003．707—714
[102] Bethke AD. Genetic Algorithms as Function Optimizers. Dissertation Abstracts International, 1981, 41(9), 3503(B).
[103] Goldberg D E. Construction of high-order deceptive functions using low-order Walsh coeffcients. Annals of Mathematics and Artificial Intelligence, 1992(5). 35-48.
[104] Deb K, Goldberg D E. Sufficient Conditions for Deceptive and Easy Binary Functions. Evolutionary Computation, 2(3), 1992. 279-311
[105] Pelikan M, Goldberg D E. Escaping Hierarchical Traps with Competent Genetic Algorithms. IlliGAL Report No. 2001003, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2001.
[106] Sastry K, Pelikan M, Goldberg D E. Combining Competent Crossover and Mutation Operators: A Probabilistic Model Building Approach. IlliGAL Report No. 2004010, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2004.
[107] Lima C F, Sastry K, Goldberg D ,E Lobo F G. Combining Competent Crossover and Mutation Operators: A Probabilistic Model Building Approach. IlliGAL Report No. 2005002, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2005.
[108] 李敏强,寇纪淞．遗传算法的模式欺骗性分析．中国科学(E辑),Vol．32(1),2002． 95—102
[109] Whitley L D. Fundamental principles of deception in genetic search, In Foundations of Genetic algorithms, Rawlins G(ed.), San Mateo, CA: Morgan Kaufmann, 1991:221-241.
[110] Homaifar A, Qi X, Fost J. Analysis and design of a general GA deceptive problem. In Proceedings of the Fourth International Conference on Genetic Algorithms(ICGA4), Belew R, Brooker L.(eds.), San Francisco: Morgan Kaufmann Publishers, Inc., 1991. 196-203
[111] Wright A H, Jonathan E R, James R N. Analysis of the Simple Genetic Algorithm on the Single-peak and Double-peak Landscapes Proceedings of CEC 2002, IEEE Press.
[112] Harik G. Linkage Learning via Probabilistic Modeling in the ECGA. IlliGAL Report No. 99010, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 1999.
[113] Chen Y-P, Goldberg D E. Convergence Time for the Linkage Learning Genetic Algorithm. IlliGAL Report No. 2003025, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2003.
[114] Sastry K, Pelikan M, Goldberg D E. Efficiency Enhancement of Genetic Algorithms via Building-Block-Wise Fitness Estimation. IlliGAL Report No. 20040010, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2004.
[115] Yu T L, Goldberg D E. Quality and Efficiency of Model Building for Genetic Algorithms. IlliGAL Report No. 2004004, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2004.
[116] Chen Ying-ping. Extending the Scalability of Linkage Learning Genetic Algorithms: Theory and Practice. IlliGAL Report No. 2004018, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2004.
[117] Sastry K, Pelikan M, Goldberg D E. Efficiency Enhancement of Probabilistic Model Building Genetic Algorithms. IlliGAL Report No. 20040020, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2004.
[118] Butz M V, Pelikan M, Xavier Llora, Goldberg D E. Extracted Global Structure Makes Local Building Block Processing Effiective in XCS. IlliGAL Report No. 20050011, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2005.
[119] Butz M V, Pelikan M, Xavier Llora, Goldberg D E. Automated Global Structure Extraction For Effiective Local Building Block Processing in XCS. IlliGAL Report No. 20050010, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2005.
[120] Yu T-L, Sastry K, Goldberg D E. Linkage Learning, Overlapping Building Blocks, and Systemmatic Strategy for Scalable Recombination. IlliGAL Report No. 20050016, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 2005.
[121] Deb K, Goldberg D E. MGA in C: A Messy Genetic Algorithm in C. IlliGAL Report No. 91008, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 1991.
[122] Munetomo M, Goldberg D E. Identifying Linkage by Nonlinearity Check. IlliGAL Report No. 98012, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 1998.
[123] Munetomo M, Goldberg D E. Designing a genetic algorithm using the identifying linkage by nonlinearity check. IlliGAL Report No. 98014, University of Illinois at Urbana-Champaign, Illinois Genetic Algorithms Laboratory, Urbana, 1998.
[124] Schaffer J D, Morishima A. An adaptive crossover distribution mechanism for genetic algorithms. Proceedings of the Second International Conference on Genetic Algorithms(ICGA-87), 1987. 36-40
[125] Levenick J R. Metabits: Genetic endogenous crossover control. Proceedings of the sixth International Conference on Genetic Algorithms(ICGA-95), 1995. 88-95
[126] Smith J, Fogarty T C. An adaptive poly-parental recombination strategy. Proceedings of AISB-95 Workshop on Evolutionary computing, 1995. 48-61
[127] Smith J, Fogarty T C. Recombination strategy adaptation via evolution of gene linkage. Proceedings of the 1996 IEEE International Conference on Evolutionary Computation, 1996. 826-831
[128] Baluja S. Population-based incremental learning: A method for integrating genetic search based function optimization and competitive learning. Tech. Rep. No. CMU-CS-94-163. Pittsburg, PA: Carnegie Mellon University.
[129] Harik G R, Lobo F G, Goldberg D E. The compact genetic algorithm. Proceedings of the 1998 IEEE International Conference on Evolutionary Computation,1998.523-528
[130]Harik G R,Lobo F G,Goldberg D E.The compact genetic algorithm.IEEE Transactions on Evolutionary Computation,3(4),287-297.
[131]Pelikan M,Goldberg D E,Cantu-Paz E.Linkage problem,distribution estimation,and Bayesian networks.Evolutionary Computation,8(3),311-341.
[132]Yu T-L,Goldberg D E,Yassine A,Chen Y-P.A Genetic Algorithm Design Inspired by Organizational Theory:A Pilot Study of a Dependency Structure Matrix Driven Genetic Algorithm.IlliGAL Report No.2003007,University of Illinois at Urbana-Champaign,Illinois Genetic Algorithms Laboratory,Urbana,2003.
[133]Yu T-L.A Matrix Approach for Finding Extrema:Problems with Modularity Hierarchy and Overlap.IlliGAL Report No.2007012,University of Illinois at Urbana-Champaign,Illinois Genetic Algorithms Laboratory,Urbana,2007.
[134]钟开莱著,刘文译.概率论教程.上海:上海科学技术出版社,1989年.
[135]施仁杰.马尔科夫链基础及其应用.西安,西安电子科技大学出版社,1992.
[136]Rudolph G.Finite Markov Chain Results in Evolutionary Computation:A Tour d'Horizon.Fundamenta Informaticae archive,Vol.35,1998.67-89
[137]Vose M D.The simple genetic algorithms:Foundations and theory.Cambridge,Massachusetts:The MIT Press,1999.
[138]Jong K A D,Spears W M,Gordon D F.Using markov chains to analyze GAFOs.Foundations of genetic algorithms 3,Whitley L D,Vose M D(eds.),San Mateo.CA:Morgan Kaufmann,1995.
[139]Rudolph G.Convergence rates of evolutionary algorithms for a class of convex objective functions,Control and Cybernetics,1997,26(3),375-390.
[140]Mahfoud S W,Goldberg D E.Parallel recombinative simulated annealing:A genetic algorithm.Parallel Computing,1995,21(1),1-28.
[141]张讲社,徐宗本,梁怡.整体退火遗传算法及其收敛充要条件.中国科学(E辑),1997,Vol.27(2),154-164.
[142]Cerf R.The dynamics of mutation-selection algorithms with large population sizes.Annales de l'Institut Henri Poincare,Probabilites et Statistiques,32(4),1996.778-817
[143]Suzuki J.A further result on the markov chain model of genetic algorithm and its application to a simulated annealing-like strategy.IEEE Transactions on Systems,Man,and Cybernetics-Part B,28(1),1998.95-102
[144]Lozano J A,Larranaga P,Grana M,Albizuri F X.Genetic algorithms:Bridging the Convergence gap.Theoretical Computer Science,229(1),1999.11-22
[145]He J,Kang K.On the convergence rates of genetic algorithms.Theoretical Computer Science,229(1),1999.23-34
[146]Vose M D,Liepins G E.Punctuated equilibria in Genetic search.Complex Systems,1991,Vol(5),3-144.
[147]Liepins G E,Vose M D.Characterizing Crossover in Genetic Algorithms.Annals of Mathematics and Artificial Intelligence,1992,5(1),27-34.
[148]Vose M D,Wright A H.Simple Genetic Algorithms with Linear Fitness,Evolutionary Computation,1994,2(4),347-368.
[149]Vose M D.Modeling simple genetic algorithms.Evolutionary Computation,1996,3,453-472.
[150]Vose M D.Random Heuristic Search,Theor.Comput.Sci.1999,229(1),103-142.
[151]Rowe J E,Vose M D,Wright A H.Group Properties of Crossover and Mutation.Evolutionary Computation,2002,10,151-184.
[152]Wright A H,Rowe J E,Poli R,C R Stephens.A Fixed Point Analysis of a Gene Pool GA with Mutation.The Genetic and Evolutionary Computation Conference (GECCO-2002),Langdon W B,et al(eds.).San Francisco:Morgan Kaufmann:2002.642-649
[153]Wright A H.Bistability of the Needle Function in the Presence of Truncation Selection.Springer-Verlag Lecture Notes in Computer Science,2004,LNCS3103,330-342.
[154]Shukla P K,Deb K.On finding multiple Pareto-optimal solutions using classical and evolutionary generating methods.European Journal of Operational Research,2007,181(3),1630-1652.
[155]Wright A H et al.Bistability in a Gene Pool GA with Mutation.In:Kenneth A,De Jong(Ed.) Foundations of Genetic Algorithms 7,Terremolinos Spain.CA:Morgan Kaufmann,2003.63-80
[156]Vose M D,Wright A H.The Simple Genetic Algorithm and the Walsh Transform:Part Ⅰ,Theory,Evolutionary Computation,1998,6(3),253-273.
[157]Vose M D,Wright A H.The Simple Genetic Algorithm and the Walsh Transform:Part Ⅱ,The Inverse,Evolutionary Computation,1998,6(3),275-289.
[158]姚妙新,陈芳启.非线性理论数学基础.天津:天津大学出版社,2005,71-92.
[159]Geiringer H.On the Probability of Linkage in Mendelian Heredity.Annals of Mathematical Statistics 15,1944.25-57
[160]IWAMATSU M.Multi-specie particle swarm optimizer for multimodal function optimization.IEICE Transactions on Information and Systems,2006,E89-D(3),1181-1187.
[161]姬振豫.正交设计的方法与理论.香港:世界科技出版社,2001.
[162]郝翔,李人厚.一种快速有效的多模态函数寻优方法—双群体遗传算法.控制理论与应用,1997,10(5),765-768.
[163]Hastings A.Stable cycling in discrete-time genetic models.Proc.Nat.Acad.Sci.USA1981.7224-7225
[164]Wright A H,Agapie A.Cyclic and Chaotic Behavior in Genetic Algorithms.The Genetic and Evolutionary Computation Conference GECCO-2001,Goodman E K,et al(eds.).Morgan Kaufmann:2001.718-724
[165]Liepins G E,Vose M D.Deceptiveness and genetic algorithms dynamics.In Rawlins G J E(Ed.) Foundations Of Genetic Algorithms 1,San Mateo,CA:Morgan Kaufmann Publishers,Inc.1991.36-50
[166]Beyer,H.G On the Dynamics of EAs without Selection.Foundations of Genetic Algorithms 5,Wolfgang Banzhaf and Colin Reeves,San Francisco:Morgan Kaufmann Press,1999.5-26
[167]Juliany J,Vose M D.The genetic algorithm fractal.Evolutionary Computation,1994,2(2),165-180.
[168]Collard P,Clergue M.Genetic Algorithm:from Hegemony to Chaos.Complex Systems,12(1),2000.1-29
[169]龙运佳.混沌工程学.中国工程科学,2001年,Vol.3(2),1-7.
[170]陈士华,陆君安.混沌动力学.武汉:武汉水利电力大学出版社,1999.
[171]王杰智,陈增强,袁著祉.一类新的混沌系统及其性质研究.物理学报,Vol.55(8),2006.3956-3963
[172]Wang Jie-Zhi,Chen Zeng-Qiang,Yuan Zhu-Zhi.The generation of a hyperchaotic system based on three-dimensional autonomous chaotic system.Chinese Physics,Vol.15(6),2006.1216-1225