基于量子机制与组合方法的智能优化算法及应用研究
|
摘要
智能优化算法,如人工神经网络、遗传算法、模拟退火、粒子群算法及其优化策略等,通过模拟或揭示某些自然现象或过程而得到发展,其内容涉及数学、物理学、生物进化、人工智能、神经科学和统计力学等方面,为解决复杂问题提供了新的思路和手段。量子计算利用了量子理论中有关量子态的叠加、纠缠和干涉等特性,通过量子并行计算使得某些在经典计算机上计算复杂度很高的问题有可能降低其复杂度。由于量子计算机尚未实现,因此如何在传统计算机利用量子计算也是一个研究的热点。本文将量子机制引入到智能优化算法中,提出几种改进的优化算法,希望利用量子机制的优点,进一步提高优化算法的求解能力及运算效率。具体内容如下:
(1)将量子机制与进化算法结合,根据旅行商问题的特点,尝试将量子进化算法应用于求解小规模旅行商问题,实验结果表明量子进化算法求解旅行商问题的可行性。
(2)提出一种量子群进化算法模型,引入量子角的概念,将量子机制与粒子群算法相结合,用于求解背包问题,得到了令人满意的结果。
(3)提出一种量子组合优化算法模型,将量子进化算法同智能优化算法结合,求解两类组合优化问题。
(4)根据可满足性问题的特点,进一步改进量子组合优化算法,并应用于求解3-SAT问题。
(5)提出了AFTER_PSO算法,并且将其应用于股票及太阳黑子数据的预测问题上,结果表明了算法的可行性与有效性。
本文的研究结果丰富了智能优化算法的研究,提高了智能优化算法求解一类优化问题的效率,同时为如何在传统计算机上应用量子机制研究提供了一种参考。
Optimization technique is based on mathematics and used to find the optimum solutions of many engineering problems. Since 1980’s, the industry control has been developing in the direction of large-scale, sequential, compositive and intricate process. Owing to the feature of complexity, constraint, nonlinearity, multi-local minimum and difficulty to model, optimality calculation of all kinds of process control is an urgent problem that needs to be solved. So it has become a primary research direction in the related subject and field to research intelligent algorithms for solving large-scale parallel problems.
Intelligent optimal algorithm is an important branch of artificial intelligent field. With the flourishing development, the research of intelligent computation is very active. The intelligent computation is also called“soft computing”. It is developed by simulating or revealing some phenomenon and process of nature. Nature is the source of human creativity. For recent years, the research of intelligent optimization algorithms is currently a focus in the advanced field. These algorithms include artificial neural network (ANN), genetic algorithm (GA), simulated annealing (SA), particle swarm optimization (PSO), ant colony optimization (ACO), artificial immune system (AIS), etc., which are developed by simulating or revealing some phenomenon and process of nature. They provide innovative thought and means to solve many complicated problems. These algorithms enrich the optimization technique and provide workable solutions for some traditional combinatorial optimization problems.
Quantum computation was proposed by Benioff and Feynman in the early 1980s. Quantum algorithms exploit the laws of quantum mechanics in order to perform efficient computation. Due to its unique computational performance, the quantum computation has attracted extensive attention of researchers. The characteristic of quantum world has made the quantum system break the limit of classical information system. Quantum computer can easily break existing cryptosystem and lead the revolution of the cryptosystem. It will influence all the departments of defense and the whole society.
Based on the intelligent optimal algorithms, quantum mechanism is introduced, some improvements are made and different algorithms are combined. The major contents could be summarized as follows:
(1) Quantum-inspired evolutionary algorithm (QEA) is the combination of quantum mechanism and evolutionary algorithm. It is based on the concept and principles of quantum computation, such as quantum bit and superposition of states. The chromosomes are coded by qubit. This probabilistic presentation can make one chromosome to present information of several states. The quantum gate is adopted to evolve superposition states. It can keep the diversity of population and avoid the pressure of choice. The information of current best individual can lead mutation which makes the population evolve to excellent schema with a large probability. According to the characteristic of travelling salesman problem, the QEA is adopted to solve small scale travelling salesman problem. The results show that using the QEA to solve travelling salesman problem is feasible.
(2) Based on quantum evolutionary algorithm and particle swarm optimization, a novel quantum swarm evolutionary algorithm (QSE) is presented. A new definition of Q-bit expression called quantum angle is proposed, and an improved particle swarm optimization is employed to update the quantum angles automatically. The simulated results in solving 0–1 knapsack problem show that QSE is superior to traditional QEA. In addition, the comparison experiments show that QSE is better than many traditional heuristic algorithms, such as climb hill algorithm, simulation anneal algorithm and taboo search algorithm.
(3) A quantum evolutionary combinatorial algorithm is presented based on the quantum-inspired evolutionary algorithm and conventional evolutionary algorithms. There are two phases and two different groups GroupI and GroupII in the proposed algorithm. In the first-phase, GroupI with population size n is optimized by the quantum-inspired evolutionary algorithm. And the GroupII with population size 2n is optimized in the second-phase. In each generation, there are n individuals of the GroupII derived from GroupI. Meanwhile, the best solution from GroupII is used to guide the update of a quarter of individuals of the GroupI. To demonstrate the effectiveness and applicability of the proposed approach, several experiments are performed on function optimization and 0-1 knapsack problems. The compared results with the quantum-inspired evolutionary algorithm show that the proposed quantum combinatorial optimal algorithm is more powerful in the aspects of population diversity, convergence speed and the ability of global optimization.
(4) The satisfiability problem (SAT) is the basic NP-hard problem in the field of computer science. As one of the six basic core NP-complete problems, it is the basic for many theoretical problems, for example, symbolic logic, reasoning, machine learning. Besides its theoretical importance, SAT has a large number of practical applications such as VLSI test and verification, design of asynchronous circuits, sports planning and so on. In this paper, we propose the improved quantum-inspired evolutionary algorithm and, extending the application field of the QEA, use it to solve the 3-SAT problem. The quantum angle is adopted in the improved algorithm. And a new quantum rotation gate strategy is adopted to update the population. Furthermore, to accelerate the convergence speed, the particle swarm optimization is combined with the IQEA. Experimental results show that the improved algorithm is feasible and effective for solving 3-SAT problems. It can be as a new incomplete method to solve satisfiability problems.
(5) A modified particle swarm optimization (PSO) algorithm is proposed. Linear constraints in the PSO are added to satisfy normalization conditions for different problems. A hybrid algorithm based on the modified PSO and aggregated forecast through exponential re-weighting (AFTER) is presented. The AFTER is adopted to initialize the weight of different problem. Then the improved PSO is adopted to optimize the population. The AFTER accelerate the search speed of PSO and the global and local search ability of PSO avoid the possibility of local optimal of Bayesian probability. Combining forecasting can improve the forecasting accuracy through combining different forecasting methods. The effectiveness of the algorithm is demonstrated through the prediction on the sunspots and the stocks data. Simulated results show that the hybrid algorithm can improve the forecasting accuracy to a great extent.
The research of this article has enriched the study of intelligent optimal algorithm and developed the efficiency of solving combinational problems. And it also provides a reference on how to apply quantum mechanism to classical computer.
(1)将量子机制与进化算法结合,根据旅行商问题的特点,尝试将量子进化算法应用于求解小规模旅行商问题,实验结果表明量子进化算法求解旅行商问题的可行性。
(2)提出一种量子群进化算法模型,引入量子角的概念,将量子机制与粒子群算法相结合,用于求解背包问题,得到了令人满意的结果。
(3)提出一种量子组合优化算法模型,将量子进化算法同智能优化算法结合,求解两类组合优化问题。
(4)根据可满足性问题的特点,进一步改进量子组合优化算法,并应用于求解3-SAT问题。
(5)提出了AFTER_PSO算法,并且将其应用于股票及太阳黑子数据的预测问题上,结果表明了算法的可行性与有效性。
本文的研究结果丰富了智能优化算法的研究,提高了智能优化算法求解一类优化问题的效率,同时为如何在传统计算机上应用量子机制研究提供了一种参考。
Optimization technique is based on mathematics and used to find the optimum solutions of many engineering problems. Since 1980’s, the industry control has been developing in the direction of large-scale, sequential, compositive and intricate process. Owing to the feature of complexity, constraint, nonlinearity, multi-local minimum and difficulty to model, optimality calculation of all kinds of process control is an urgent problem that needs to be solved. So it has become a primary research direction in the related subject and field to research intelligent algorithms for solving large-scale parallel problems.
Intelligent optimal algorithm is an important branch of artificial intelligent field. With the flourishing development, the research of intelligent computation is very active. The intelligent computation is also called“soft computing”. It is developed by simulating or revealing some phenomenon and process of nature. Nature is the source of human creativity. For recent years, the research of intelligent optimization algorithms is currently a focus in the advanced field. These algorithms include artificial neural network (ANN), genetic algorithm (GA), simulated annealing (SA), particle swarm optimization (PSO), ant colony optimization (ACO), artificial immune system (AIS), etc., which are developed by simulating or revealing some phenomenon and process of nature. They provide innovative thought and means to solve many complicated problems. These algorithms enrich the optimization technique and provide workable solutions for some traditional combinatorial optimization problems.
Quantum computation was proposed by Benioff and Feynman in the early 1980s. Quantum algorithms exploit the laws of quantum mechanics in order to perform efficient computation. Due to its unique computational performance, the quantum computation has attracted extensive attention of researchers. The characteristic of quantum world has made the quantum system break the limit of classical information system. Quantum computer can easily break existing cryptosystem and lead the revolution of the cryptosystem. It will influence all the departments of defense and the whole society.
Based on the intelligent optimal algorithms, quantum mechanism is introduced, some improvements are made and different algorithms are combined. The major contents could be summarized as follows:
(1) Quantum-inspired evolutionary algorithm (QEA) is the combination of quantum mechanism and evolutionary algorithm. It is based on the concept and principles of quantum computation, such as quantum bit and superposition of states. The chromosomes are coded by qubit. This probabilistic presentation can make one chromosome to present information of several states. The quantum gate is adopted to evolve superposition states. It can keep the diversity of population and avoid the pressure of choice. The information of current best individual can lead mutation which makes the population evolve to excellent schema with a large probability. According to the characteristic of travelling salesman problem, the QEA is adopted to solve small scale travelling salesman problem. The results show that using the QEA to solve travelling salesman problem is feasible.
(2) Based on quantum evolutionary algorithm and particle swarm optimization, a novel quantum swarm evolutionary algorithm (QSE) is presented. A new definition of Q-bit expression called quantum angle is proposed, and an improved particle swarm optimization is employed to update the quantum angles automatically. The simulated results in solving 0–1 knapsack problem show that QSE is superior to traditional QEA. In addition, the comparison experiments show that QSE is better than many traditional heuristic algorithms, such as climb hill algorithm, simulation anneal algorithm and taboo search algorithm.
(3) A quantum evolutionary combinatorial algorithm is presented based on the quantum-inspired evolutionary algorithm and conventional evolutionary algorithms. There are two phases and two different groups GroupI and GroupII in the proposed algorithm. In the first-phase, GroupI with population size n is optimized by the quantum-inspired evolutionary algorithm. And the GroupII with population size 2n is optimized in the second-phase. In each generation, there are n individuals of the GroupII derived from GroupI. Meanwhile, the best solution from GroupII is used to guide the update of a quarter of individuals of the GroupI. To demonstrate the effectiveness and applicability of the proposed approach, several experiments are performed on function optimization and 0-1 knapsack problems. The compared results with the quantum-inspired evolutionary algorithm show that the proposed quantum combinatorial optimal algorithm is more powerful in the aspects of population diversity, convergence speed and the ability of global optimization.
(4) The satisfiability problem (SAT) is the basic NP-hard problem in the field of computer science. As one of the six basic core NP-complete problems, it is the basic for many theoretical problems, for example, symbolic logic, reasoning, machine learning. Besides its theoretical importance, SAT has a large number of practical applications such as VLSI test and verification, design of asynchronous circuits, sports planning and so on. In this paper, we propose the improved quantum-inspired evolutionary algorithm and, extending the application field of the QEA, use it to solve the 3-SAT problem. The quantum angle is adopted in the improved algorithm. And a new quantum rotation gate strategy is adopted to update the population. Furthermore, to accelerate the convergence speed, the particle swarm optimization is combined with the IQEA. Experimental results show that the improved algorithm is feasible and effective for solving 3-SAT problems. It can be as a new incomplete method to solve satisfiability problems.
(5) A modified particle swarm optimization (PSO) algorithm is proposed. Linear constraints in the PSO are added to satisfy normalization conditions for different problems. A hybrid algorithm based on the modified PSO and aggregated forecast through exponential re-weighting (AFTER) is presented. The AFTER is adopted to initialize the weight of different problem. Then the improved PSO is adopted to optimize the population. The AFTER accelerate the search speed of PSO and the global and local search ability of PSO avoid the possibility of local optimal of Bayesian probability. Combining forecasting can improve the forecasting accuracy through combining different forecasting methods. The effectiveness of the algorithm is demonstrated through the prediction on the sunspots and the stocks data. Simulated results show that the hybrid algorithm can improve the forecasting accuracy to a great extent.
The research of this article has enriched the study of intelligent optimal algorithm and developed the efficiency of solving combinational problems. And it also provides a reference on how to apply quantum mechanism to classical computer.
引文
[1].王凌.智能优化算法及其应用[M].北京:清华大学出版社,2001.
[2].邓先礼.最优化技术[M],重庆:重庆大学出版社,1998.
[3].刑文训,谢金星.现代优化计算方法[M],北京:清华大学出版社,2005.
[4]. Osman H.I. Metaheuristics: a bibliography[J]. Annals of Operations Research,1996,63: 513-623.
[5]. Michael A.Nielsen, Isaac L.Chuang. Quantum Computation and Quantum Information[M]. Cambridge University Press, 2000.
[6]. Benioff P. The Computer as a Physical System: a Microscopic Quantum Mechanical Hamiltonian Model of Computers as Represented by Turing Machines[J]. J. Stat. Phys., 1980, 22: 563-591.
[7]. Feynman R.P. Simulating Physics with Computers[J]. International Journal of Theoretical Physics, 1982, 21(6&7): 467-488.
[8]. Barenco A., Deutsch D., Ekert A.and Jozsa R.Conditional quantum dynamics and logic gates[J]. Phys. Rev. Lett. , 1995 , 74(20): 4083-4088.
[9]. Deutsch D , Jozsa R.Rapid solution of the problems by quantum computation. Proceedings - Royal Society, Mathematical and physical sciences,439(1907): 553-558,1992[C].
[10]. Shor P.W. Quantum Computing. Documenta Mathematica. Extra Volume, Proceedings of the International Congress of Mathematicians , 467-486,1998[C].Berlin,1998.
[11]. Grover L.K. Algorithms for Quantum Computation: Discrete Logarithms and Factoring. Proceedings of the 35th Annual Symposium on Foundations of Computer Science,124-134,1994[C]. Piscataway,NJ: IEEE Press,1994.
[12]. Deutsch D. Quantum theory,the Church-Turing principle and the universal quantum computer. Proceedings of the Royal Society, Series A,400: 97-117,1985[C].
[13]. Shor P.W. Algorithms for quantum computation: Discrete logarithms and factoring. Proc of the 35th Annual Sysp on Foundations of Computer Science,124-134,1994[C].New Mexico: IEEE Computer Society Press, 1994.
[14]. Grover L.K. A Fast Quantum Mechanical Algorithm for Database Search. Proceedings of the 28th Annual ACM Symposium on the Theory of Computing,212-221, 1996[C]. Philadelphia, Pennsylvania: ACM Press, 1996.
[15].王凌,吴昊,唐芳,郑大钟,金以慧.混合量子遗传算法及其性能分析[J].控制与决策, 2005,20(2): 156-160.
[16].周春光,梁艳春.计算智能[M].吉林:吉林大学出版社,2001.
[17].张乃尧,阎平凡.神经网络与模糊控制[M].北京:清华大学出版社,1998.
[18].丁永生.计算智能——理论、技术与应用[M].北京:科学出版社,2004.
[19]. Nirwan A. Computational Intelligence for Optimization[M].Kluwer Academic Publishers, 1997.
[20]. Fredric M.H, Ivica K. Principles of Neurocomputing for Science and Engineering[M].New York Publishers, 2001.
[21]. Taplak H,Uzmay I,Yildirim S.An artificial neural network application to fault detection of a rotor bearing system[J]. Industrial Lubrication and Tribology,2006,58(1): 32-44.
[22]. Parrilla M , Aranda J , Dormido-Canto S.Parallel evolutionary computation: Application of an EA to controller design[J]. Lecture Notes in Computer Science,2005,3562: 153-162.
[23]. Metropolis N,Rosenbluth A,Rosenbluth M,Teller A,Teller E. Equations of state calculations by fast computing machines[J]. Journal of Chemical Physics,1953,21: 1087-1091.
[24]. Kirkpartick S,Gelatt C D,Vecchi M P. Optimization by simulated annealing[J]. Science,1983,220: 671-680.
[25].王乃龙,戴宏宇,周润德,用模拟退火算法实现集成电路热布局优化[J].半导体学报,2003,24(4):203-208.
[26].黄亮,李翠华,解布局问题的并行模拟退火算法与实现技术[J].厦门大学学报(自然科学版),2003,42(1):21-28.
[27]. Aydin M.E , Fogarty T.C. A simulated annealing algorithm for multi-agent systems: a job-shop scheduling application[J]. Journal of Intelligent Manufacturing,2004,15 (6):805-814.
[28]. Wang T.Y , Wu K.B.A revised simulated annealing algorithm for obtaining the minimum total tardiness in job shop scheduling problems[J].International Journal of Systems Science, 2000, 31 (4): 537-542.
[29]. Steinhofel K,Albrecht A,Wong C.K. On various cooling schedules for simulated annealing applied to the job shop problem[J]. Lecture Notes in Computer Science,1998,1518: 260-279.
[30]. Martinez Alfaro H,Ruiz Cruz M.A. Using simulated annealing for discrete optimal control systems design[J].Lecture Notes in Artificial Intelligence,2004,3315: 932-941.
[31]. Motoda T,Stengel R.F,Miyazawa Y. Robust control system design using simulated annealing[J]. Journal of Guidance Control and Dynamics, 2002,25(2):267-274.
[32]. Kavaklioglu K, Upadhyaya B.R. Optimal fuzzy control design using simulated annealing and application to feedwater heater control[J]. Nuclear Technology,1999,125(1): 70-84.
[33].张雪江,朱向阳,钟秉林,黄仁.基于模拟退火算法的知识获取方法的研究[J].控制与决策,1997,12(4):327-331.
[34]. Chen HC,Shankaranarayanan G,She LL,Lyer A. Machine learning approach to inductive query by examples: An experiment using relevancefeedback, ID3,genetic algorithms, and simulated annealing[J]. Journal of the American Society for Information Science, 1998, 49(8): 693-705.
[35]. Feng Y.C,Cai X. Simulated annealing schemes in transiently chaotic neural network model[J]. International Journal of Modern Physics B, 2004,18(17-19): 2579-2584.
[36]. Lahiri A,Chakravorti S. A novel approach based on simulated annealing coupled to artificial neural network for 3-D electric-field optimization[J]. IEEE Transactions on Power Delivery, 2005, 20(3): 2144-2152.
[37]. Sarkar D , Modak J.M. ANNSA: a hybrid artificial neural network/simulated annealing algorithm for optimal control problems[J]. Chemical Engineering Science,2003,58(14): 3131-3142.
[38]. Zhang C.J,Wang X.D,Zhang H R. Contrast enhancement for image with simulated annealing algorithm and wavelet neural network[J]. Lecture Notes in Computer Science,2005,3497: 707-712.
[39]. Kim H.C,Boo C.J,Lee Y J. Image reconstruction using simulated annealing algorithm in EIT[J]. International Journal of Control Automation and Systems,2005,3(2): 211-216.
[40]. Gluhovsky I. Evolutionary simulated annealing with application to image restoration[J]. Journal of Computational and Graphical Statistics,2004,13(4): 871-885.
[41]. Maglogiannis I,Zafiropoulos E. Automated medical image registration using the simulated annealing algorithm[J]. Lecture Notes in Computer Science,2004,3025: 456-465.
[42]. Kennedy J.,Eberhart R.C. Particle swarm optimization. Proceedings of IEEE International Conference On neural networks,4: 1942-1948. 1995[C]. IEEE,1995.
[43]. Reynolds C. Flocks,Herds,Schools. A Distributed Behavioral Model[J]. Computer Graphics,1987,21(4): 25-34.
[44].曹春红.几何约束求解技术的研究[D].吉林:吉林大学计算机科学与技术学院,2005.
[45]. Shi Y. , Eberhart R.C. A modified particle swarm optimizer. Proceedings of the International Joint Conference on Evolutionary Computation,69-73,1998[C].
[46]. Shi Y., Eberhart R.C. Empirical study of particle swarm optimization. Proceedings of the Word Multiconference on Systemics, Cybernetics and Informatics,1945-1950,2000[C].Orlando: FL. 2000.
[47]. Mikki S, Kishk A. Improved particle swarm optimization technique using hard boundary conditions[J]. Microwave and Optical Technology Letters,2005,46(5): 422-426.
[48]. Liu B,Wang L,Jin Y.H,Tang F, Huang D.X. Improved particle swarm optimization combined with chaos[J]. Chaos Solitons Fractals,2005, 25(5): 1261-1271.
[49]. Jian W,Xue Y.C,Qian J.X. An improved particle swarm optimization algorithm with disturbance. IEEE International Conference on Systems, Man and Cybernetics, 6: 5900-5904,2004[C]. New York: IEEE, 2004.
[50]. G.W. Mackey. Quantum mechanics and Hilbert space[J]. Amer Math, 1957,64(2): 45-57.
[51]. Cohen D.W. An Introduction to Hilbert Space and Quantum Logic[M]. New York: Springer, 1989.
[52].夏培肃.量子计算[J].计算机研究与发展, 2001, 38(10): 1153-1171.
[53]. Deutsch D. Quantum computational networks[J]. Proc Roy Soc London A,1989,425: 73- 90.
[54]. Vedral V.,Barenco A., Ekert A. Quantum networks for elementary arithmetic operations[J]. Phys Rev A,1996,54(1): 147-153.
[55]. Han Kuk-Hyun and Kim Jong-Hwan. Quantum-inspired Evolutionary Algorithm for a Class of Combinatorial Optimization[J]. IEEE Transactions on Evolutionary Computation, IEEE Press, 2002, 6(6): 580-593.
[56]. Han Kuk-Hyun and Kim Jong-Hwan. Quantum-Inspired Evolutionary Algorithms With a New Termination Criterion, H?Gate, and Two-Phase Scheme[J]. IEEE Transactions on Evolutionary Computation, 2004, 8(2): 156-169.
[57]. Hey Tony. Quantum computing: An introduction[J]. Computing&Control Engineering Journal,1996,10(3): 105-112.
[58].杨俊安,庄镇泉,史亮.多宇宙并行量子遗传算法[J].电子学报, 2004, 32(6): 923-928.
[59].郭海燕,金炜东,李丽,罗碧华.分组量子遗传算法及其应用[J].西南科技大学学报, 2004, 19(1): 18-21.
[60].郭海燕.基于混沌优化的量子遗传算法[J].电子测量技术, 2006, 29(2): 14-15.
[61]. Choi I.C,Kim S.I,Kim H.S. A genetic algorithm with a mixed region search for the asymmetric traveling salesman problem[J]. Computers & Operations Research, 2003, 30 (5): 773-786.
[62]. Yoon H.S,Moon B.R. An empirical study on the synergy of multiple crossover operators[J]. IEEE Transactions on Evolutionary Computation, 2002, 6 (2): 212-223.
[63]. Tang L.X,Liu J.Y, Rong A.Y, Yang Z.H. Modelling and a genetic algorithm solution for the slab stack shuffling problem when implementing steel rolling schedules[J]. International Journal of Production Research, 2002, 40(7): 1583~1595.
[64]. Fogel D.B. Applying evolutionary programming to selected traveling salesman problems[J]. Cybernetics and Systems, 1993, 24 (1): 27~36.
[65]. Chellapilla K, Fogel D.B. Exploring self-adaptive methods to improve the efficiency of generating approximate solutions to traveling salesman problems using evolutionary programming[J]. Lecture Notesin Computer Science, 1997, 1213: 361.
[66]. Nurnberg H T, Beyer H G. Dynamics of evolution strategies in the optimization of traveling salesman problems[J]. Lecture Notes in Computer Science, 1997, 1213: 349.
[67]. L.G. Bendall: Domination Analysis Beyond the Traveling Salesman Problem[D]. Department of Mathematics, University of Kentucky, 2004.
[68]. Tang X.J. Genetic Algorithms with Application to Engineering Optimization[D]. University of Memphis, 2004.
[69]. Wang K.P, Huang L, Zhou C.G, Pang W. Particle Swarm Optimization for Traveling Salesman Problem. Proceeding of the 2nd ICMLC, 3: 1583-1585, 2003[C]. Xi’an, China: IEEE Press, 2003.
[70].吕欣,冯登国.背包问题的量子算法分析[J].北京航空航天大学学报, 2004, 30(11): 1088-1091.
[71].李肯立,李庆华,戴光明,周炎涛.背包问题的一种自适应算法[J].计算机研究与发展, 2004, 41(7): 1292-1297.
[72].李晓萌,戴光明,石红玉.解决多维0/1背包问题的遗传算法综述[J].,电脑开发与应用, 2006, 19(1): 4-8.
[73].沈显君,王伟武,郑波尽,李元香.基于改进的微粒群优化算法的0-1背包问题求解[J].计算机工程, 2006, 32(18): 23-25.
[74].熊焰,陈欢欢,苗付友,王行甫.一种解决组合优化问题的量子遗传算法QGA[J].电子学报, 2004, 32(11): 1855-1858.
[75].苏晓琴,郭光灿.量子通信与量子计算[J].量子电子学报, 2004, 21(6): 706-718.
[76].李映,张艳宁,赵荣椿.量子搜索和进化搜索算法的比较研究[J].计算机工程与应用, 2004,18:1-4.
[77].解光军,范海秋,操礼程.一种量子神经计算网络模型[J].复旦学报(自然科学版), 2004, 43(5):700-703.
[78].宋辉,戴葵,王志英.一种改进的量子搜索算法[J].计算机工程与科学, 2002, 24(5): 4-7.
[79].张葛祥,李娜,金炜东,胡来招.一种新量子遗传算法及其应用[J].电子学报, 2004, 32(3): 476-479.
[80].杨俊安,李斌,庄镇泉,钟子发.基于量子遗传算法的盲源分离算法研究[J].小型微型计算机系统,2003, 24(8):1518-1523.
[81]. Cook S.A. The complexity of theorem-proving procedures. Proceedings Third Annual ACM Symposium on Theory of Computing, 151-158,1971[C].
[82]. Michael R. Garey, David S. Johnson. Computers and Intractability, A Guide to the Theory of NP-Completeness[M]. San Francisco: W.H. Freeman & Company, 1979.
[83]. Frédéric Lardeux, Frédéric Saubion, Jin-Kao Hao. GASAT: A Genetic Local Search Algorithm for the Satisfiability Problem[J]. Evolutionary Computation, 2006, 14(2): 223-253.
[84]. Istvan Borgulya. An Evolutionary Framework for 3-SAT Problems[J]. Journal of Computing and Information Technology - CIT 11, 2003:185-191.
[85]. Martin Davis, George Logemann, and Donald Loveland. A machine program for theorem-proving[J]. Communications of the ACM, 1962, 5(7): 394–397.
[86]. Olivier Dubois, Gilles Dequen. A backbone-search heuristic for efficient solving of hard 3-SAT formulae. Proc. of the Seventeenth International Joint Conference on Artificial Intelligence (IJCAI’01), 248–253, 2001[C]. San Francisco, 2001.
[87]. Li C.M and Anbulagan Anbulagan. Heuristics based on unit propagation for satisfiability problems. Proc. of the Fifteenth International Joint Conference on Artificial Intelligence (IJCAI’97), 366–371, 1997[C].
[88]. Belaid Benhamou, Lakhdar Sais. Theoretical study of symmetries in propositional calculus and applications. CADE’92, 281–294, 1992[C].
[89]. Kenneth A. De Jong, William M. Spears. Using genetic algorithm to solve NPcomplete problems. In Proc. of the 3rd International Conference on Genetic Algorithms (ICGA’89), 124–132, 1989[C]. Virginia, 1989.
[90]. Jens Gottlieb, Elena Marchiori, Claudio Rossi. Evolutionary algorithms for the satisfiability problem. Evolutionary Computation[J]. 2002, 10(1): 35–50.
[91]. Pierre Hansen , Brigitte Jaumard. Algorithms for the maximum satisfiability problem. Computing[J]. 1990, 44(4): 279–303.
[92]. WilliamM. Spears. Simulated annealing for hard satisfiability problems[J]. Second DIMACS implementation challenge: cliques , coloring and satisfiability, volume 26 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 1996: 533–558.
[93]. Mitchell D, Selman B, Levesque H. Hard and easy distributions of SAT problems. Proceeding of the 10th National Conference on Artificial Intelligence , American Association for Artificial Intelligence, 459-465, 1992[C].
[94]. Jens Gottlieb and Nico Voss. Adaptive fitness functions for the satisfiability problem. Parallel Problem Solving from Nature - PPSN VI 6th International Conference, 621-630, 2000[C]. France: LNCS, 2000.
[95].凌应标,吴向军,姜云飞.基于子句权重学习的求解SAT问题的遗传算法[J].计算机学报. 2005, 28(29): 1476-1482.
[96]. Benchmark. 2000. http://www.satlib.org/.
[97].张有为.预测的数学方法[M].北京:国防工业出版社,1991.
[98].熊大国.随机过程理论与应用[M].北京:国防工业出版社,1991.
[99].黎子良.统计、推断、决策[M].天津:南开大学出版社,1988.
[100]. Bates J.N and Granger C.W.J. The combination of forecasts[J]. Operations research quarterly. 1969, 20: 319-325.
[101]. Lachetermacher G, Fuller J.D. Back propagation in time series forecasting[J]. Journal of Forecasting, 1995, 14: 381-393.
[102]. Schwerzel R, Rosen B. Improving the accuracy of financial time series prediction using ensemble networks and high order statistics[J]. IEEE International Conference on Neural networks, 1997, 4: 2045-2050.
[103]. Zhang X. Time series analysis and prediction by neural networks[J]. Optimization Methods and software, 1994, 4: 151-170.
[104].李敏强,徐博艺,宼纪淞.遗传算法与神经网络的结合[J].系统工程理论与实践,1999,19(2):65-69.
[105]. Holden K, Peel D, A Unbiasedness, efficiency and the combination of economic forecasts[J]. Forecasting, 1989, 8(3):175-188.
[106]. Yang Y. Combining time series models for forecasting[J]. International Journal of Forecasting. 2004, 20(1): 69-84.
[107]. Yang Y. Combining different procedures for adaptive regression[J]. Journal of Multivariate Analysis, 2000, 74: 135-161.
[108].王硕,唐小我,曾勇.基于加速遗传算法的组合预测方法研究[J].科研管理, 2002, 23(3): 118-121.
[109].杨桂元,唐小我.提高组合预测模型精度的方法探讨[J].预测, 1997, 1: 43-45.
[110].孙延风.金融时间序列预测的神经网络方法[D].吉林:吉林大学, 2003.
[111]. Kuo R.J. A decision support system for the stock market through integration of fuzzy neural networks and fuzzy Delphi[J]. Applied Intelligence,1998, 6: 501-520.
[2].邓先礼.最优化技术[M],重庆:重庆大学出版社,1998.
[3].刑文训,谢金星.现代优化计算方法[M],北京:清华大学出版社,2005.
[4]. Osman H.I. Metaheuristics: a bibliography[J]. Annals of Operations Research,1996,63: 513-623.
[5]. Michael A.Nielsen, Isaac L.Chuang. Quantum Computation and Quantum Information[M]. Cambridge University Press, 2000.
[6]. Benioff P. The Computer as a Physical System: a Microscopic Quantum Mechanical Hamiltonian Model of Computers as Represented by Turing Machines[J]. J. Stat. Phys., 1980, 22: 563-591.
[7]. Feynman R.P. Simulating Physics with Computers[J]. International Journal of Theoretical Physics, 1982, 21(6&7): 467-488.
[8]. Barenco A., Deutsch D., Ekert A.and Jozsa R.Conditional quantum dynamics and logic gates[J]. Phys. Rev. Lett. , 1995 , 74(20): 4083-4088.
[9]. Deutsch D , Jozsa R.Rapid solution of the problems by quantum computation. Proceedings - Royal Society, Mathematical and physical sciences,439(1907): 553-558,1992[C].
[10]. Shor P.W. Quantum Computing. Documenta Mathematica. Extra Volume, Proceedings of the International Congress of Mathematicians , 467-486,1998[C].Berlin,1998.
[11]. Grover L.K. Algorithms for Quantum Computation: Discrete Logarithms and Factoring. Proceedings of the 35th Annual Symposium on Foundations of Computer Science,124-134,1994[C]. Piscataway,NJ: IEEE Press,1994.
[12]. Deutsch D. Quantum theory,the Church-Turing principle and the universal quantum computer. Proceedings of the Royal Society, Series A,400: 97-117,1985[C].
[13]. Shor P.W. Algorithms for quantum computation: Discrete logarithms and factoring. Proc of the 35th Annual Sysp on Foundations of Computer Science,124-134,1994[C].New Mexico: IEEE Computer Society Press, 1994.
[14]. Grover L.K. A Fast Quantum Mechanical Algorithm for Database Search. Proceedings of the 28th Annual ACM Symposium on the Theory of Computing,212-221, 1996[C]. Philadelphia, Pennsylvania: ACM Press, 1996.
[15].王凌,吴昊,唐芳,郑大钟,金以慧.混合量子遗传算法及其性能分析[J].控制与决策, 2005,20(2): 156-160.
[16].周春光,梁艳春.计算智能[M].吉林:吉林大学出版社,2001.
[17].张乃尧,阎平凡.神经网络与模糊控制[M].北京:清华大学出版社,1998.
[18].丁永生.计算智能——理论、技术与应用[M].北京:科学出版社,2004.
[19]. Nirwan A. Computational Intelligence for Optimization[M].Kluwer Academic Publishers, 1997.
[20]. Fredric M.H, Ivica K. Principles of Neurocomputing for Science and Engineering[M].New York Publishers, 2001.
[21]. Taplak H,Uzmay I,Yildirim S.An artificial neural network application to fault detection of a rotor bearing system[J]. Industrial Lubrication and Tribology,2006,58(1): 32-44.
[22]. Parrilla M , Aranda J , Dormido-Canto S.Parallel evolutionary computation: Application of an EA to controller design[J]. Lecture Notes in Computer Science,2005,3562: 153-162.
[23]. Metropolis N,Rosenbluth A,Rosenbluth M,Teller A,Teller E. Equations of state calculations by fast computing machines[J]. Journal of Chemical Physics,1953,21: 1087-1091.
[24]. Kirkpartick S,Gelatt C D,Vecchi M P. Optimization by simulated annealing[J]. Science,1983,220: 671-680.
[25].王乃龙,戴宏宇,周润德,用模拟退火算法实现集成电路热布局优化[J].半导体学报,2003,24(4):203-208.
[26].黄亮,李翠华,解布局问题的并行模拟退火算法与实现技术[J].厦门大学学报(自然科学版),2003,42(1):21-28.
[27]. Aydin M.E , Fogarty T.C. A simulated annealing algorithm for multi-agent systems: a job-shop scheduling application[J]. Journal of Intelligent Manufacturing,2004,15 (6):805-814.
[28]. Wang T.Y , Wu K.B.A revised simulated annealing algorithm for obtaining the minimum total tardiness in job shop scheduling problems[J].International Journal of Systems Science, 2000, 31 (4): 537-542.
[29]. Steinhofel K,Albrecht A,Wong C.K. On various cooling schedules for simulated annealing applied to the job shop problem[J]. Lecture Notes in Computer Science,1998,1518: 260-279.
[30]. Martinez Alfaro H,Ruiz Cruz M.A. Using simulated annealing for discrete optimal control systems design[J].Lecture Notes in Artificial Intelligence,2004,3315: 932-941.
[31]. Motoda T,Stengel R.F,Miyazawa Y. Robust control system design using simulated annealing[J]. Journal of Guidance Control and Dynamics, 2002,25(2):267-274.
[32]. Kavaklioglu K, Upadhyaya B.R. Optimal fuzzy control design using simulated annealing and application to feedwater heater control[J]. Nuclear Technology,1999,125(1): 70-84.
[33].张雪江,朱向阳,钟秉林,黄仁.基于模拟退火算法的知识获取方法的研究[J].控制与决策,1997,12(4):327-331.
[34]. Chen HC,Shankaranarayanan G,She LL,Lyer A. Machine learning approach to inductive query by examples: An experiment using relevancefeedback, ID3,genetic algorithms, and simulated annealing[J]. Journal of the American Society for Information Science, 1998, 49(8): 693-705.
[35]. Feng Y.C,Cai X. Simulated annealing schemes in transiently chaotic neural network model[J]. International Journal of Modern Physics B, 2004,18(17-19): 2579-2584.
[36]. Lahiri A,Chakravorti S. A novel approach based on simulated annealing coupled to artificial neural network for 3-D electric-field optimization[J]. IEEE Transactions on Power Delivery, 2005, 20(3): 2144-2152.
[37]. Sarkar D , Modak J.M. ANNSA: a hybrid artificial neural network/simulated annealing algorithm for optimal control problems[J]. Chemical Engineering Science,2003,58(14): 3131-3142.
[38]. Zhang C.J,Wang X.D,Zhang H R. Contrast enhancement for image with simulated annealing algorithm and wavelet neural network[J]. Lecture Notes in Computer Science,2005,3497: 707-712.
[39]. Kim H.C,Boo C.J,Lee Y J. Image reconstruction using simulated annealing algorithm in EIT[J]. International Journal of Control Automation and Systems,2005,3(2): 211-216.
[40]. Gluhovsky I. Evolutionary simulated annealing with application to image restoration[J]. Journal of Computational and Graphical Statistics,2004,13(4): 871-885.
[41]. Maglogiannis I,Zafiropoulos E. Automated medical image registration using the simulated annealing algorithm[J]. Lecture Notes in Computer Science,2004,3025: 456-465.
[42]. Kennedy J.,Eberhart R.C. Particle swarm optimization. Proceedings of IEEE International Conference On neural networks,4: 1942-1948. 1995[C]. IEEE,1995.
[43]. Reynolds C. Flocks,Herds,Schools. A Distributed Behavioral Model[J]. Computer Graphics,1987,21(4): 25-34.
[44].曹春红.几何约束求解技术的研究[D].吉林:吉林大学计算机科学与技术学院,2005.
[45]. Shi Y. , Eberhart R.C. A modified particle swarm optimizer. Proceedings of the International Joint Conference on Evolutionary Computation,69-73,1998[C].
[46]. Shi Y., Eberhart R.C. Empirical study of particle swarm optimization. Proceedings of the Word Multiconference on Systemics, Cybernetics and Informatics,1945-1950,2000[C].Orlando: FL. 2000.
[47]. Mikki S, Kishk A. Improved particle swarm optimization technique using hard boundary conditions[J]. Microwave and Optical Technology Letters,2005,46(5): 422-426.
[48]. Liu B,Wang L,Jin Y.H,Tang F, Huang D.X. Improved particle swarm optimization combined with chaos[J]. Chaos Solitons Fractals,2005, 25(5): 1261-1271.
[49]. Jian W,Xue Y.C,Qian J.X. An improved particle swarm optimization algorithm with disturbance. IEEE International Conference on Systems, Man and Cybernetics, 6: 5900-5904,2004[C]. New York: IEEE, 2004.
[50]. G.W. Mackey. Quantum mechanics and Hilbert space[J]. Amer Math, 1957,64(2): 45-57.
[51]. Cohen D.W. An Introduction to Hilbert Space and Quantum Logic[M]. New York: Springer, 1989.
[52].夏培肃.量子计算[J].计算机研究与发展, 2001, 38(10): 1153-1171.
[53]. Deutsch D. Quantum computational networks[J]. Proc Roy Soc London A,1989,425: 73- 90.
[54]. Vedral V.,Barenco A., Ekert A. Quantum networks for elementary arithmetic operations[J]. Phys Rev A,1996,54(1): 147-153.
[55]. Han Kuk-Hyun and Kim Jong-Hwan. Quantum-inspired Evolutionary Algorithm for a Class of Combinatorial Optimization[J]. IEEE Transactions on Evolutionary Computation, IEEE Press, 2002, 6(6): 580-593.
[56]. Han Kuk-Hyun and Kim Jong-Hwan. Quantum-Inspired Evolutionary Algorithms With a New Termination Criterion, H?Gate, and Two-Phase Scheme[J]. IEEE Transactions on Evolutionary Computation, 2004, 8(2): 156-169.
[57]. Hey Tony. Quantum computing: An introduction[J]. Computing&Control Engineering Journal,1996,10(3): 105-112.
[58].杨俊安,庄镇泉,史亮.多宇宙并行量子遗传算法[J].电子学报, 2004, 32(6): 923-928.
[59].郭海燕,金炜东,李丽,罗碧华.分组量子遗传算法及其应用[J].西南科技大学学报, 2004, 19(1): 18-21.
[60].郭海燕.基于混沌优化的量子遗传算法[J].电子测量技术, 2006, 29(2): 14-15.
[61]. Choi I.C,Kim S.I,Kim H.S. A genetic algorithm with a mixed region search for the asymmetric traveling salesman problem[J]. Computers & Operations Research, 2003, 30 (5): 773-786.
[62]. Yoon H.S,Moon B.R. An empirical study on the synergy of multiple crossover operators[J]. IEEE Transactions on Evolutionary Computation, 2002, 6 (2): 212-223.
[63]. Tang L.X,Liu J.Y, Rong A.Y, Yang Z.H. Modelling and a genetic algorithm solution for the slab stack shuffling problem when implementing steel rolling schedules[J]. International Journal of Production Research, 2002, 40(7): 1583~1595.
[64]. Fogel D.B. Applying evolutionary programming to selected traveling salesman problems[J]. Cybernetics and Systems, 1993, 24 (1): 27~36.
[65]. Chellapilla K, Fogel D.B. Exploring self-adaptive methods to improve the efficiency of generating approximate solutions to traveling salesman problems using evolutionary programming[J]. Lecture Notesin Computer Science, 1997, 1213: 361.
[66]. Nurnberg H T, Beyer H G. Dynamics of evolution strategies in the optimization of traveling salesman problems[J]. Lecture Notes in Computer Science, 1997, 1213: 349.
[67]. L.G. Bendall: Domination Analysis Beyond the Traveling Salesman Problem[D]. Department of Mathematics, University of Kentucky, 2004.
[68]. Tang X.J. Genetic Algorithms with Application to Engineering Optimization[D]. University of Memphis, 2004.
[69]. Wang K.P, Huang L, Zhou C.G, Pang W. Particle Swarm Optimization for Traveling Salesman Problem. Proceeding of the 2nd ICMLC, 3: 1583-1585, 2003[C]. Xi’an, China: IEEE Press, 2003.
[70].吕欣,冯登国.背包问题的量子算法分析[J].北京航空航天大学学报, 2004, 30(11): 1088-1091.
[71].李肯立,李庆华,戴光明,周炎涛.背包问题的一种自适应算法[J].计算机研究与发展, 2004, 41(7): 1292-1297.
[72].李晓萌,戴光明,石红玉.解决多维0/1背包问题的遗传算法综述[J].,电脑开发与应用, 2006, 19(1): 4-8.
[73].沈显君,王伟武,郑波尽,李元香.基于改进的微粒群优化算法的0-1背包问题求解[J].计算机工程, 2006, 32(18): 23-25.
[74].熊焰,陈欢欢,苗付友,王行甫.一种解决组合优化问题的量子遗传算法QGA[J].电子学报, 2004, 32(11): 1855-1858.
[75].苏晓琴,郭光灿.量子通信与量子计算[J].量子电子学报, 2004, 21(6): 706-718.
[76].李映,张艳宁,赵荣椿.量子搜索和进化搜索算法的比较研究[J].计算机工程与应用, 2004,18:1-4.
[77].解光军,范海秋,操礼程.一种量子神经计算网络模型[J].复旦学报(自然科学版), 2004, 43(5):700-703.
[78].宋辉,戴葵,王志英.一种改进的量子搜索算法[J].计算机工程与科学, 2002, 24(5): 4-7.
[79].张葛祥,李娜,金炜东,胡来招.一种新量子遗传算法及其应用[J].电子学报, 2004, 32(3): 476-479.
[80].杨俊安,李斌,庄镇泉,钟子发.基于量子遗传算法的盲源分离算法研究[J].小型微型计算机系统,2003, 24(8):1518-1523.
[81]. Cook S.A. The complexity of theorem-proving procedures. Proceedings Third Annual ACM Symposium on Theory of Computing, 151-158,1971[C].
[82]. Michael R. Garey, David S. Johnson. Computers and Intractability, A Guide to the Theory of NP-Completeness[M]. San Francisco: W.H. Freeman & Company, 1979.
[83]. Frédéric Lardeux, Frédéric Saubion, Jin-Kao Hao. GASAT: A Genetic Local Search Algorithm for the Satisfiability Problem[J]. Evolutionary Computation, 2006, 14(2): 223-253.
[84]. Istvan Borgulya. An Evolutionary Framework for 3-SAT Problems[J]. Journal of Computing and Information Technology - CIT 11, 2003:185-191.
[85]. Martin Davis, George Logemann, and Donald Loveland. A machine program for theorem-proving[J]. Communications of the ACM, 1962, 5(7): 394–397.
[86]. Olivier Dubois, Gilles Dequen. A backbone-search heuristic for efficient solving of hard 3-SAT formulae. Proc. of the Seventeenth International Joint Conference on Artificial Intelligence (IJCAI’01), 248–253, 2001[C]. San Francisco, 2001.
[87]. Li C.M and Anbulagan Anbulagan. Heuristics based on unit propagation for satisfiability problems. Proc. of the Fifteenth International Joint Conference on Artificial Intelligence (IJCAI’97), 366–371, 1997[C].
[88]. Belaid Benhamou, Lakhdar Sais. Theoretical study of symmetries in propositional calculus and applications. CADE’92, 281–294, 1992[C].
[89]. Kenneth A. De Jong, William M. Spears. Using genetic algorithm to solve NPcomplete problems. In Proc. of the 3rd International Conference on Genetic Algorithms (ICGA’89), 124–132, 1989[C]. Virginia, 1989.
[90]. Jens Gottlieb, Elena Marchiori, Claudio Rossi. Evolutionary algorithms for the satisfiability problem. Evolutionary Computation[J]. 2002, 10(1): 35–50.
[91]. Pierre Hansen , Brigitte Jaumard. Algorithms for the maximum satisfiability problem. Computing[J]. 1990, 44(4): 279–303.
[92]. WilliamM. Spears. Simulated annealing for hard satisfiability problems[J]. Second DIMACS implementation challenge: cliques , coloring and satisfiability, volume 26 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 1996: 533–558.
[93]. Mitchell D, Selman B, Levesque H. Hard and easy distributions of SAT problems. Proceeding of the 10th National Conference on Artificial Intelligence , American Association for Artificial Intelligence, 459-465, 1992[C].
[94]. Jens Gottlieb and Nico Voss. Adaptive fitness functions for the satisfiability problem. Parallel Problem Solving from Nature - PPSN VI 6th International Conference, 621-630, 2000[C]. France: LNCS, 2000.
[95].凌应标,吴向军,姜云飞.基于子句权重学习的求解SAT问题的遗传算法[J].计算机学报. 2005, 28(29): 1476-1482.
[96]. Benchmark. 2000. http://www.satlib.org/.
[97].张有为.预测的数学方法[M].北京:国防工业出版社,1991.
[98].熊大国.随机过程理论与应用[M].北京:国防工业出版社,1991.
[99].黎子良.统计、推断、决策[M].天津:南开大学出版社,1988.
[100]. Bates J.N and Granger C.W.J. The combination of forecasts[J]. Operations research quarterly. 1969, 20: 319-325.
[101]. Lachetermacher G, Fuller J.D. Back propagation in time series forecasting[J]. Journal of Forecasting, 1995, 14: 381-393.
[102]. Schwerzel R, Rosen B. Improving the accuracy of financial time series prediction using ensemble networks and high order statistics[J]. IEEE International Conference on Neural networks, 1997, 4: 2045-2050.
[103]. Zhang X. Time series analysis and prediction by neural networks[J]. Optimization Methods and software, 1994, 4: 151-170.
[104].李敏强,徐博艺,宼纪淞.遗传算法与神经网络的结合[J].系统工程理论与实践,1999,19(2):65-69.
[105]. Holden K, Peel D, A Unbiasedness, efficiency and the combination of economic forecasts[J]. Forecasting, 1989, 8(3):175-188.
[106]. Yang Y. Combining time series models for forecasting[J]. International Journal of Forecasting. 2004, 20(1): 69-84.
[107]. Yang Y. Combining different procedures for adaptive regression[J]. Journal of Multivariate Analysis, 2000, 74: 135-161.
[108].王硕,唐小我,曾勇.基于加速遗传算法的组合预测方法研究[J].科研管理, 2002, 23(3): 118-121.
[109].杨桂元,唐小我.提高组合预测模型精度的方法探讨[J].预测, 1997, 1: 43-45.
[110].孙延风.金融时间序列预测的神经网络方法[D].吉林:吉林大学, 2003.
[111]. Kuo R.J. A decision support system for the stock market through integration of fuzzy neural networks and fuzzy Delphi[J]. Applied Intelligence,1998, 6: 501-520.