基于光学原理的最优化方法研究
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摘要
生活中,实际工程中的优化问题存在着高维数、大计算量、多局部极优值等复杂特点。这使得传统优化算法的求解无法达到实际应用要求。所以研究新的优化方法以解决此类复杂的优化问题具有重要的实际意义。
自然界的优化发展方式给复杂的实际工程优化问题的解决提供了新的思路和方法。自然界的美丽源于其以简单的规则描述的复杂万物,而在这个描述的过程中,自然界总是按照某种形式的最优(最节省)去发展,从而体现其经济本性。本文旨在提出一种通过模拟自然现象,不需要设计太多的经验性的参数,并且没有随机因素在迭代搜索过程中的优化算法。
作为最普通的自然现象之一,光线在不均匀介质中的传播也体现了一种能量的节省。本文受启发于此现象,在较详细的分析了光线传播的折射,反射定律之后,根据费马原理,提出了一种新的智能优化算法——光线寻优算法(Light Ray Optimization)。光线寻优算法是一种通过模拟光线在不均匀介质中的传播过程进行最优值搜索的寻优算法。算法以网格划分优化问题的可行域,并赋予不同网格不同的介质密度以改变在其中光线的传播速度。这使得光线在不同的网格之间传播时发生折射现象和反射现象,并以此不断地在不同网格中更新搜索方向和搜索位置,并使算法最终自动收敛到目标函数的最优值。本文阐述了选取矩形网格的必要性,分析了光线的搜索方向的更新过程。本文在二维光线寻优算法的基础上,分析了光线寻优算法在n(n>2)维空间中搜索的可行性,讨论了在n维空间中光线与搜索子空间之间角度的定义方式以及光线搜索方向和搜索位置的更新过程,并将光线寻优算法推广到n维空间中。
本文对光线寻优算法寻优的收敛性进行了分析,以定理的形式证明了光线会在水平和竖直方向偏向最优值所在的方向,并通过在水平,竖直方向上不定期交替地发生折射现象更新搜索方向,使光线寻优算法收敛到局部的最优值。
在数值实验中,将光线寻优算法应用于几个各具特点的标准测试函数中。通过光线寻优算法在不同维数空间中对标准测试函数的搜索结果,说明了光线寻优算法的搜索特点和对不同类优化问题的搜索可行性。根据经典测试函数的不足,引入随机测试函数,对光线寻优算法的搜索性能的进行一步评判,进而讨论光线寻优算法的可行性及有效性。
通过分析光线寻优算法对经典测试函数的搜索结果,本文对如何缩短光线寻优算法的搜索时间和如何搜索有负值的优化问题进行了讨论。本文讨论了搜索二维空间中的最优值的意义,并在二维光线寻优算法的基础上,通过采用具有良好性质的正六边形网格划分可行域,提高光线寻优算法的搜索效率。对于在n维空间中的光线寻优算法,本文通过初始设置较大的网格,并在迭代搜索过程中,以不同形式不断地减小网格尺寸,缩短搜索时间和提高搜索精度。本文通过对目标函数进行的不同变换,使目标函数符合光线寻优算法的使用要求,使光线寻优算法可以求解有负值的优化问题,从而扩大光线寻优算法的应用范围。
本文还将光线寻优算法与群智能优化算法的代表粒子群优化算法进行了比较。从搜索机制和搜索过程两方面分析光线寻优算法与已存在的主要优化方法相比,分析了光线寻优算法所存在的优势与不足。本文在已有工作的基础上,提出了光线寻优算法在未来包括参数设置,搜索方向的更新方式,多路搜索等主要的改进方向。
The optimization problems in real life and engineering have the complex features, such as high dimensions, huge calculations, multi optima and so on. It makes the traditional optimization methods can not meet the practical requirements. So, researching a new optimization method to tackle the complex problem has a significant practical meaning.
The optimized development pattern of nature provides a new solution to deal with the complex characters in the engineering optimization problems. The beauty of the nature comes from its depicting of all creations with simple rules. In this process, the nature shows out the economic with certain pattern of the optimal development.
This paper will propose a new optimization method, in which there is no empirical parameter and no random elements. As one of the commonest phenomena in nature, the light ray propagating in uneven medium shows out the saving of energy. Inspired from this phenomenon, a new optimization method--Light Ray Optimization, according to the Fermat's Principle is proposed after analyzing the refraction and the reflection of light ray. Light Ray Optimization (LRO for short) is an iterative optimization method which searches the optimization by simulating the light ray propagating in uneven transparent medium. The LRO divides the feasible region with mesh to get grids. Each grid has its medium density. Because of the different fitness value which leads different speed of light ray, the light ray will refract or reflect when propagating between the different grids, and update the searching position and searching direction. Finally, the LRO will convergent to the optimization of the problem automatically. This paper illustrates the importance of the grid chosen, and analyzes the updating process of searching direction. Based on the LRO in the2-dimensional space, the possibility of the LRO searching in n-dimensional space(n>2) is analyzed. The updating of searching direction and the definition of the angle in n-dimensional space is discussed. And the LRO is expended into the n-dimensional space.
The convergence of LRO is analyzed. The theorems proof that the searching direction of light ray will lean to the steepest descent direction in horizontal and vertical direction. With the shift between the horizontal and vertical searching direction updating, LRO will convergent to the local optimization.
In the numerical experiment, the LRO is applied on several benchmarks with different characters. After analyzing the searching results of the benchmarks in different dimensional spaces via LRO, the features and feasibility of LRO to different optimization problems are given. Because of the shortage of classical benchmarks, the random benchmark is introduced into the paper to test the searching ability of LRO. And the features and feasibility of LRO are further discussed.
From analyzing the searching results of traditional benchmarks in numerical experiment, the paper furtherly discussed how to reduce the searching time of LRO and how to search the optimization problem with negative value. The paper discussed the practical meaning of LRO searching in2-dimensional space, and based on the2-dimensional LRO, the improvement of the grid is given. The hexagonal grid with good property is introduced to improve the searching efficiency. To LRO in N-dimensional space, the initialized big size grid is given. And with the iterative searching going on, the grid size shrinks with different patterns to reduce the searching time and improve the searching precision. This paper applied several transforms to the objective function with negative value. The modified objective function meets the requirement of LRO. The feasibility of LRO is extended.
This paper compares the LRO and the PSO, which is a class of swarm intelligence. Both the advantage and the disadvantage are illustrated by analyzing the searching mechanism and searching process. The results are testified by the numerical experiment. Based on the works above, the future work of LRO is proposed, such as the initialized parameters of LRO, the searching direction updating pattern, and multi-start searching and so on.
自然界的优化发展方式给复杂的实际工程优化问题的解决提供了新的思路和方法。自然界的美丽源于其以简单的规则描述的复杂万物,而在这个描述的过程中,自然界总是按照某种形式的最优(最节省)去发展,从而体现其经济本性。本文旨在提出一种通过模拟自然现象,不需要设计太多的经验性的参数,并且没有随机因素在迭代搜索过程中的优化算法。
作为最普通的自然现象之一,光线在不均匀介质中的传播也体现了一种能量的节省。本文受启发于此现象,在较详细的分析了光线传播的折射,反射定律之后,根据费马原理,提出了一种新的智能优化算法——光线寻优算法(Light Ray Optimization)。光线寻优算法是一种通过模拟光线在不均匀介质中的传播过程进行最优值搜索的寻优算法。算法以网格划分优化问题的可行域,并赋予不同网格不同的介质密度以改变在其中光线的传播速度。这使得光线在不同的网格之间传播时发生折射现象和反射现象,并以此不断地在不同网格中更新搜索方向和搜索位置,并使算法最终自动收敛到目标函数的最优值。本文阐述了选取矩形网格的必要性,分析了光线的搜索方向的更新过程。本文在二维光线寻优算法的基础上,分析了光线寻优算法在n(n>2)维空间中搜索的可行性,讨论了在n维空间中光线与搜索子空间之间角度的定义方式以及光线搜索方向和搜索位置的更新过程,并将光线寻优算法推广到n维空间中。
本文对光线寻优算法寻优的收敛性进行了分析,以定理的形式证明了光线会在水平和竖直方向偏向最优值所在的方向,并通过在水平,竖直方向上不定期交替地发生折射现象更新搜索方向,使光线寻优算法收敛到局部的最优值。
在数值实验中,将光线寻优算法应用于几个各具特点的标准测试函数中。通过光线寻优算法在不同维数空间中对标准测试函数的搜索结果,说明了光线寻优算法的搜索特点和对不同类优化问题的搜索可行性。根据经典测试函数的不足,引入随机测试函数,对光线寻优算法的搜索性能的进行一步评判,进而讨论光线寻优算法的可行性及有效性。
通过分析光线寻优算法对经典测试函数的搜索结果,本文对如何缩短光线寻优算法的搜索时间和如何搜索有负值的优化问题进行了讨论。本文讨论了搜索二维空间中的最优值的意义,并在二维光线寻优算法的基础上,通过采用具有良好性质的正六边形网格划分可行域,提高光线寻优算法的搜索效率。对于在n维空间中的光线寻优算法,本文通过初始设置较大的网格,并在迭代搜索过程中,以不同形式不断地减小网格尺寸,缩短搜索时间和提高搜索精度。本文通过对目标函数进行的不同变换,使目标函数符合光线寻优算法的使用要求,使光线寻优算法可以求解有负值的优化问题,从而扩大光线寻优算法的应用范围。
本文还将光线寻优算法与群智能优化算法的代表粒子群优化算法进行了比较。从搜索机制和搜索过程两方面分析光线寻优算法与已存在的主要优化方法相比,分析了光线寻优算法所存在的优势与不足。本文在已有工作的基础上,提出了光线寻优算法在未来包括参数设置,搜索方向的更新方式,多路搜索等主要的改进方向。
The optimization problems in real life and engineering have the complex features, such as high dimensions, huge calculations, multi optima and so on. It makes the traditional optimization methods can not meet the practical requirements. So, researching a new optimization method to tackle the complex problem has a significant practical meaning.
The optimized development pattern of nature provides a new solution to deal with the complex characters in the engineering optimization problems. The beauty of the nature comes from its depicting of all creations with simple rules. In this process, the nature shows out the economic with certain pattern of the optimal development.
This paper will propose a new optimization method, in which there is no empirical parameter and no random elements. As one of the commonest phenomena in nature, the light ray propagating in uneven medium shows out the saving of energy. Inspired from this phenomenon, a new optimization method--Light Ray Optimization, according to the Fermat's Principle is proposed after analyzing the refraction and the reflection of light ray. Light Ray Optimization (LRO for short) is an iterative optimization method which searches the optimization by simulating the light ray propagating in uneven transparent medium. The LRO divides the feasible region with mesh to get grids. Each grid has its medium density. Because of the different fitness value which leads different speed of light ray, the light ray will refract or reflect when propagating between the different grids, and update the searching position and searching direction. Finally, the LRO will convergent to the optimization of the problem automatically. This paper illustrates the importance of the grid chosen, and analyzes the updating process of searching direction. Based on the LRO in the2-dimensional space, the possibility of the LRO searching in n-dimensional space(n>2) is analyzed. The updating of searching direction and the definition of the angle in n-dimensional space is discussed. And the LRO is expended into the n-dimensional space.
The convergence of LRO is analyzed. The theorems proof that the searching direction of light ray will lean to the steepest descent direction in horizontal and vertical direction. With the shift between the horizontal and vertical searching direction updating, LRO will convergent to the local optimization.
In the numerical experiment, the LRO is applied on several benchmarks with different characters. After analyzing the searching results of the benchmarks in different dimensional spaces via LRO, the features and feasibility of LRO to different optimization problems are given. Because of the shortage of classical benchmarks, the random benchmark is introduced into the paper to test the searching ability of LRO. And the features and feasibility of LRO are further discussed.
From analyzing the searching results of traditional benchmarks in numerical experiment, the paper furtherly discussed how to reduce the searching time of LRO and how to search the optimization problem with negative value. The paper discussed the practical meaning of LRO searching in2-dimensional space, and based on the2-dimensional LRO, the improvement of the grid is given. The hexagonal grid with good property is introduced to improve the searching efficiency. To LRO in N-dimensional space, the initialized big size grid is given. And with the iterative searching going on, the grid size shrinks with different patterns to reduce the searching time and improve the searching precision. This paper applied several transforms to the objective function with negative value. The modified objective function meets the requirement of LRO. The feasibility of LRO is extended.
This paper compares the LRO and the PSO, which is a class of swarm intelligence. Both the advantage and the disadvantage are illustrated by analyzing the searching mechanism and searching process. The results are testified by the numerical experiment. Based on the works above, the future work of LRO is proposed, such as the initialized parameters of LRO, the searching direction updating pattern, and multi-start searching and so on.
引文
[1]Glover F, Gutin G, Yeo A, Zverovich A. Construction heuristics for the asymmetric TSP[J]. European Journal of Operational Research, 2001,129(3):555-568P
[2]Chou C M, Su K T, Tseng C H. Effectiveness of street sweeping and washing for controlling ambient TSP[J]. Atmospheric Environment, 2005,39(10):1891-1902P
[3]Engebretsen L, Karpinski M. TSP with bounded metrics[J]. Journal of Computer and System Sciences,2006,72(4):509-546P
[4]Wang J P, Hu M. A solution for TSP based on artificial fish algorithm[C]. Proceedings of the 2009 International Conference on Computational Intelligence and Natural Computing,2009. Wuhan,2009: 26-29P
[5]肖东喜,朱金福.飞机排班中航班环的动态构建方法[J].系统工程.2007,25(11)：19-25页
[6]刘家学,郑昌义,刘耀武.带有约束的运输问题及其推广应用[J].系统工程理论与实践.2002,22(2)：127-130页
[7]Krishnan P. Automatic synthesis of schedulers in timed systems[J]. Electronic Notes in Theoretical Computer Science,2000,31:118-131P
[8]叶建芳,王正肖,潘晓弘.免疫粒子群优化算法在车间作业调度中的应用[J].浙江大学学报.2008,42(5)：863-868页
[9]韩世莲,刘新旺.多目标多模式模糊运输问题的最优折衷解[J].系统工程.2007,25(9)：26-32页
[10]温强,胡明明,桑楠.基于彩色线阵CCD的大米色选算法[J].农业机械学报.2008,39(10)：105-108页
[11]陈孝敬,吴迪,何勇.基于小波包和偏最小二乘支持向量机的多光谱纹理图像的大米分类研究[J].光谱学与光潜分析.2009,29(1)：222-225页
[12]万仲平,费浦生编著.优化理论与方法[M].第一版.武汉：武汉大学出版 社,2004：10-62页
[13]李春明编著.优化方法[M].第一版.南京：东南大学出版社,2009：10-50页
[14]Eiben A E. Genetic algorithms with multi-parent recombination[C]. Proceedings of the International Conference on Evolutionary Computation,1994. Orlando,1994:78-87P
[15]Syswerda, G. Uniform crossover in genetic algorithms[C]. Proceedings of the Third International Conference on Genetic Algorithms, George mason University,1989. San Francisco:Morgan Kaufmann Publishers Inc,1989:2-9P
[16]Wu A, Tsang P W, Yuen T Y, Yeung L F. Affine invariant object shape matching using genetic algorithm with multi-parent orthogonal recombination and migrant principle[J]. Applied Soft Computing Journal,2009,9(1):282-289P
[17]元昌安,彭昱忠等编著.基因表达式编程原理与应用[M].第一版.北京：科学出版社,2010：1-170页
[18]玄光男,程润伟,于歆杰等编著.遗传算法与工程优化[M].第·版.北京：清华大学出版社,2004：1-52页
[19]姜大立,杨西龙,杜文.车辆路径问题的遗传算法研究[J].系统工程理论与实践.1999,19(6)：40-45页
[20]曾相戈,韩伯棠.一种求解带资源约束的并行机器多目标调度问题的遗传算法[J].系统工程理论与实践.2005,25(9)：78-82页
[21]韩丽霞,王宇平.解旅行商问题的一个新的遗传算法[J].系统工程理论与实践.2007,27(12)：145-150页
[22]胡大伟,陈诚.遗传算法(GA)和禁忌搜索算法(TS)在配送中心选址和路线问题中的应用[J].系统工程理论与实践.2007,27(9)：171-176页
[23]许项东,程琳.城市道路单行系统布局优化的双层规划模型和混合算法[J].系统工程理论与实践.2009,29(10)：180-187页
[24]江中央,蔡自兴,王勇.求解全局优化问题的混合自适应正交遗传算法[J].软件学报.2010,21(6)：1296-1307贞
[25]Farasat A, Menhaj M B, Mansouri T, Moghadam M R S. A new model-free optimization algorithm inspired from asexual reproduction[J]. Applied Soft Computing Journal,2010,10(4):1284-1292P
[26]Kirkpatrick S C D, Gelatt M P, Vecchi. Optimization by Simulated Annealing[J]. Science,1983,220(4598):671-680P
[27]Cerny V. A thermodynamical approach to the travelling salesman problem:an efficient simulation algorithm[J]. Journal of Optimization Theory and Applications,1985,45:41-51P
[28]Granville V, Krivanek J P, Rasson. Simulated annealing:A proof of convergence[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence,1994,16(6):652-656P
[29]Vicente J D, Lanchares J, Hermida R. Placement by Thermodynamic Simulated Annealing[J]. Physics Letters A,2003,317(5):415-423P
[30]王银年,葛洪伟.求解TSP问题的改进模拟退火遗传算法[J].计算机工程与应用.2010,46(5)：44-47页
[31]杨卫波,赵燕伟.求解TSP问题的改进模拟退火算法[J].计算机工程与应用.2010,46(15)：34-36页
[32]许小勇.混沌模拟退火算法在数值函数优化中的应用[J].计算机与数字工程.2010,38(3)：37-40页
[33]冯洪奎,鲍劲松,金烨.混合粒子群优化算法求解多车辆拖动货物问题[J].计算机集成制造系统.2010,16(7)：1427-1436页
[34]许智宏,宋勃,郭艳艳.模拟退火与蚁群混合并行算法解旅行商问题[J].河北工业大学学报.2010,39(2)：48-51页
[35]肖思和,鲁红英,范安东.模拟退火算法在求解组合优化问题中的应用研究[J].四川理工学院学报.2010,23(1)：1296-1307页
[36]Glover F. Tabu Search-Part Ⅰ[J]. ORSA Journal on Computing,1989, 1 (3):190-206P
[37]Glover F. Tabu Search-Part Ⅱ[J]. ORSA Journal on Computing,1990, 2(1):4-32P
[38]Cvijovic D, Klinowski J. Taboo search-an approach to the multiple minima problem[J]. Science,1995,267:664-666P
[39]董宗然,陈明华,李迎秋.最短路径问题的禁忌搜索求解方法[J].计算机工程与应用.2010,46(33)：36-38页
[40]李俊青,潘全科,王玉亭.多目标柔性车间调度的Pareto混合禁忌搜索算法[J].计算机集成制造系统.2010,16(7)：1419-1426页
[41]刘霞,齐欢.基于禁忌搜索的动态车辆路径问题研究[J].武汉理工大学学报.2010,34(2)：293-296页
[42]张芬,谢安世,周传华.适于高维空间搜索的自组织学习算法[J].计算机工程与设计.2010,31(9)：2005-2009页
[43]张汉强,卢建刚,陈金水.新的混合智能优化算法及其多目标优化应用[J].计算机应用.2010,30(5)：1290-1292页
[44]Zong W G, Joong H K, Loganathan G V. A New Heuristic Optimization Algorithm:Harmony Search[J]. Simulation,2001,76(2):60-68P
[45]李亮,迟世春,林皋.改进和声搜索算法及其在土坡稳定分析中的应用[J].土木工程学报.2006,39(5)：107-111页
[46]金永强,苏怀智,李子阳.基于和声搜索的边坡稳定性投影寻踪聚类分析[J].水利学报.2007,1：682-686页
[47]Marco D, Thomas S编著.蚁群优化[M].第一版.北京：清华大学出版社,2007：1-112页
[48]Dorigo M. Optimization, Learning and Natural Algorithms [D]. Italie Politecnico di Milano,1992:45-47P
[49]Dorigo M, Maniezzo V. Colorni, Ant system:optimization by a colony of cooperating agents[J]. IEEE Transactions on Systems, Man, and Cybernetics-Part B,1996,26(1):29-41P
[50]Dorigo M, Gamhardella L M. Ant Colony System:A Cooperative Learning Approach to the Traveling Salesman Problem[J]. IEEE Transactions on Evolutionary Computation,1997,1(1):53-66P
[51]Caro G D, Dorigo M. AntNet:a mobile agents approach to adaptive routing[C]. Proceedings of the Thirty-First Hawaii International Conference on System Science,1998:74-83P
[52]Bseten M, Stutzle T, Dorigo, M. Ant colony optimization for the total weighted tardiness prohlem[J]. Lecture Notes in Computer Science, 2000,1917:611-620P
[53]刘利强.蚁群优化方法研究及其在潜艇导航规划中的应用[D].哈尔滨工程 大学博士学位论文.2008：1-60页
[54]http://demonstrations.wolfram.com/FoodSearchingModelForAnts/
[55]Parpinelli R S, Lopes H S, Freitas A. Data mining with an ant colony optimization algorithm[J]. IEEE Transaction on Evolutionary Computation,2002,6(2):321-332P
[56]Martens D, Backer D M, Haesen R, Vanthienen J. Classification with Ant Colony Optimization[J]. IEEE Transactions on Evolutionary Computation,2007,11(5):651-665P
[57]Donati A V, Montemanni R, Casagrande N, Rizzoli A E, Gambardella L M. Time Dependent Vehicle Routing Problem with a Multi Ant Colony System[J]. European Journal of Operational Research,2008,185(3): 1174-1191P
[58]Chen P W, Zhang J. Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements[J]. IEEE Transactions on Systems, Man, and Cybernetics-Part C:Applications and Reviews,2009,31(1):29-43P
[59]Abbaspour K C, Schulin R, Van G M T. Estimating unsaturated soil hydraulic parameters using ant colony optimization[J]. Advances In Water Resources,2001,24(8):827-841P
[60]赵书强,王磊.改进蚁群在配电网规划中的应用[J].电力系统保护与控制.2010,24：61-65页
[61]李丽,牛奔等编著.粒子群优化算法[M].第一版.北京：冶金工业出版社,2009：1-70页
[62]Balas E, Vazacopoulos A. Guided local search with shifting bottleneck for job shop scheduling[J]. Management Science,1998, 44(2):262-274P
[63]Voudouris C, Tsang E. Guided local search with shifting bottleneck for job shop scheduling[J]. European Journal of Operational Research,1999,113(2):469-499P
[64]Flores L, Gilberto, Reed, Martin J. Guided local search as a network planning algorithm that incorporates uncertain traffic demands [J]. Computer Networks,2007,51(11):3172-3196P
[65]Hu G, Xiong W, Jiang B, Yuan J, Zhang X. Binary ant colony algorithm with balanced search bias[C]. Proceedings of the World Congress on Intelligent Control and Automation,2010:3120-3125P
[66]Lara A, Sanchez G, Coello C A C, Schutze 0. A new local search strategy for memetic multiobjective evolutionary algorithms [J]. IEEE Transactions on Evolutionary Computation,2010,14(1): 112-132P
[67]李兰英,张雷雷,石敏.改进的二维增强贪婪软硬件划分算法[J].计算机工程与应用.2009,45(21)：64-67页
[68]刘星宝,蔡自兴,王勇.用于全局优化问题的混合免疫进化算法[J].西安电子科技大学学报.2010,37(5)：971-980页
[69]燕乐纬,陈树辉,李森.一种改进的广义遗传算法及其在鲁棒优化问题中的应用[J].振动与冲击.2010,29(4)：30-33页
[70]王子若,陈永昌编著.优化计算方法[M].第一版.北京：机械工业出版社,1989：1-42页
[71]沈继红.基于光学原理的优化方法的研究[C].哈尔滨工程大学理学院学术年会.黑龙江哈尔滨,2007.哈尔滨：哈尔滨工程大学,2007：20-21页
[72]Shen J H, Li Y. Optimization Algorithm Based on Optical Principles[C].国际一般系统研究会中国分会,Harbin University of Science and Technology,2007. Harbin:Harbin University of Science and Technology Pr,2007:20-24P
[73]Shen J H, Li Y. An optimization algorithm based on optical princ.iples[J]. Advances in Systems Science and Applications,2009, 9(4):647-655P
[74]Shen J H, Li Y, Light Ray Optimization and its parameters analysis[C]. 2009 International Joint Conference on computational sciences and optimization, Sanya, Hainan.2009,2:918-922P
[75]沈继红,李焱.基于正六边形网格的光线寻优算法[C].中国运筹学会成立三十周年暨2010学术年会,2010,北京.2010：21-22页
[76]Shen J H, Li J L. The principle analysis of Light Ray Optimization Algorithm[C].2010 Second International Conference on Computational Intelligence and Natural Computing Proceedings. CINC, China,2010,2:154-157P
[77]Shen J H, Li Y. Light Ray Optimization with Function Transform[C]. The 8th International conference on Optimization: Techniques and Applications. Shanghai.2010:21-22P
[78]梁柱编著.光学原理教程[M].第一版.北京：北京航空航天大学出版社,2005：1-70页
[79]盛国.推动波动光学发展的几个重要实验[J].物理通报.2007,9(1)：50-52页
[80]单天明.从波动说和微粒说之争谈假说的科学功能[J].学理论.2011,4：43-44页
[81]Y. H. Li, Energics and Philosophy [M]. ShanXi Science and Technology Press. Taiyuan.2005:1-30P
[82]Richard S. Optical Refraction and Fermat's Principle at a Point [J]. Journal of the Optical Society of America.1950,40:244-245P
[83]塔拉.最小作用量原理与简单性原则[J].内蒙古大学学报.2003,35(1)：116-120页
[84]张慧鹏.物理学中的一个原理一最小作用量原理[J].物理通报.2009,5：16-18页
[85]Tan K, Liang C, Shi X. Optic eikonal Fermat's principle and the least action principle[J]. Science in China Series G:Physics, Mechanics and Astronomy.2008.51:1826-1834P
[86]Ville R. I Kaila, Arto Annila. Natural selection for least act ion [J]. Proceedings of the royal society A.2008.464:3055-3070P
[87]http://en.wikipedia.org/wiki/speed_of_light.
[88]吴佩萱.光传播中的极值问题[J].物理与工程.2005,15(5)：21-23页
[89]谢申监,周积义译.多维空间画法几何及其应用[M]北京：清华大学出版社,1983：1-10页
[90]Xu L Y, Shen J H, Bi X J. A novel particle swarm optimizer based on space contraction [J]. Journal of Harbin Engineering Unviersity, 2006,27:542-546P
[91]Winston P H. Artificial Intelligence[M] 3rd edition, Addison-Welley, Reading MA,1992:1-20P
[92]Ali T, Al A, Azzedine Z. A new modified particle swarm optimization algorithm for adaptive equal ization[J]. Digital Signal Processing, 2011,21(2):195-207P
[93]Ali A, Atai. On the limitations of classical benchmark functions for evaluating robustness of evolutionary algorithms[J]. Applied Mathematics and Computation,2010,215:3222-3229P
[94]Gaviano M, Lera D. Test. functions wi th variable attraction regions or global optimization problems[J]. Journal of global Optimization, 1998,13:207-223P
[95]Ahrari A, Shariat M, Atai A. GEM:a novel evolutionary optimization method with improved neighbourhood search[J]. Appl. Math. Comput, 2009,210:376-386P
[96]Yang G Y, Guang Y. Particle swarm optimization with mutation for best position of particles[J]. Journal of Harbin Engineering University,2006,27:531-536P
[97]Zhang H, Zhang T N, Shen J H. Research on decision-makings of structure optimization based on improved Tent PSO[J]. control and Decision,2008,23(8):857-862P
[98]Gao F, Cui G. Novel multi-step position-selectable updating particle swarm optimization algori thm[J]. Acta Electronica Sinica, 2009,37(3):529-534P
[99]Li H, He X, Xie X L. A new boundary condition for particle swarm optimization[J]. Journal of Convergence Information Technology, 2010,5(9):21-22P
[100]Ghorng Y S, Yi P H. Efficient hexagonal inner search for fast motion estimation[C]. International Conference on Image Processing,2005, Genova, Italy.2005:1093-1096P
[101]Quan E, Lalush D S. Evaluation of hexagonal and square geometries for motion-free arrayed-source X-ray micro-CT[C].2007 4th IEEE International Symposium on Biomedical Imaging:Macro to Nano,2007, Arlington, VA, USA.2007:221-224P
[102]梁森,陈花玲.常见蜂窝胞元轴向承载能力研究[J].四川兵工学报.2011,32(1)：65-69页
[103]Crypton. The mathematics of the honeycomb[J]. Science Digest.1985, 93 (6):74-77P
[104]Jie J, Zeng J C, Han C Z, Wang Q H. Knowledge-based cooperative particle swarm optimization[J]. Applied Mathematics and Computation,2008,205(2):861-873P
[105].Selleri S, Mussetta M, Pirinoli P, Zich R. Some insight over new variations of the particle swarm optimization method[J]. IEEEE Atennas and wireless propagationletters,2006,5(1):235-238P
[106]LiuXY, Liu H, Duan H C. Particle swarm optimization based on dynamic niche technology with applications to conceptual design[J]. Advances in Engineering Software,2007,38(10):668-676P
[107]Jiang M, Yang S Y. Stochastic convergence analysis and parameter selection of the standard particle swarm optimization algorithm[J]. Information Processing Letters,2007,102(1):8-16P
[108]Carlisle A, Doziert G. An off-the-shelf PSO[C]. Proceedings of the Workshop on Particle Swarm Optimization. Indianapolis, USA,2001: 1-6P
[109]曾建潮,介婧,崔志华.微粒群算法[M].北京：科学出版社,2004：10-11,104-120页
[110]张丽平,俞欢军,陈德钊,胡上序.粒子群优化算法的分析与改进[J].信息与控制.2004,33(5)：513-517页
[111]傅阳光,周成平,丁明跃.标准粒子群优化算法的收敛性分析[J].应用数学.2011,24(1)：187-194页
[112]Kennedy J, Eberhart R. Particle swarm optimization[C]. IEEE International Conference on Neural Networks-Conference Proceedings,1995:1942-1948P
[113]Kennedy J, Eberhart R. New optimizer using particle swarm theory[C]. Proceedings of the International Symposium on Micromechatronics and Human Science,1995:39-43P
[114]Shi Y, Eberhart R. Modified particle swarm optimizer[C]. Proceedings of the TEEE Conference on Evolutionary Computation, ICEC,1998:69-73P
[115]Reeves W T. Particle systems-a technique for modeling a class of fuzzy objects [J]. ACM Transactions on Graphics,1983,2(2):91-108P
[116]闵克学,葛宏伟,张毅,梁艳春.基于蚁群和粒子优化的混合算法求解TSP问题[J].吉林大学学报.2006,24(4)：402-405页
[117]柴宝杰,刘大为.基于粒子群优化的蚁群算法在TSP中的应用[J].计算机仿真.2009,26(8)：89-91页
[118]郑洁,李凯,李晓.用于求解对称旅行商问题的粒子群算法和蚂蚁算法的融合[J].计算机应用与软件.2010,27(1)：224-227页
[119]李宁,孙德宝,岑翼刚,邹彤.带变异算子的粒子群优化算法[J].计算机工程与应用.2004,(17)：12-14页
[120]张炯,刘天琪,苏鹏.基于遗传粒子群混合算法的机组组合优化[J].电力系统保护与控制.2009,37(9)：25-29页
[121]Shi Y H, Eberhart R C. Parameter selection in particle swarm optimization[C]. Proceedings of the Seventh Annual Conference on Evolutionary Programming. New York,1998:591-600P
[122]Shi Y H, Eberhart R C. Empirical study of particle swarm optimization[J]. Proceedings of the IEEE Congress on Evolutionary Computation. Piscataway, NJ:IEEE ServiceCenter,1999:1945-1950P
[123]Clerc M. The swarm and the queen:towards a deterministic and adaptive particle swarm optimization[C]. Proceedings of the IEEE Congress on Evolutionary Computation. IEEE Press,1999.1951-1957P
[124]杨维,李歧强.粒子群优化算法综述[J].中国工程科学.2004,5：87-94页
[125]纪震,周家锐,廖惠连等.智能单粒子优化算法[J].计算机学报.2010,33(3)：556-561页
[126]孙传峰,周刘喜.粒子群优化中最大速度选择[J].计算机仿真.2007,24(5)：162-164页
[127]Clerc M, Kennedy J. The particle swarm-explosion, stability, and convergence in a multidimensional complex space[J]. IEEE Transactions on Evolutionary Computation,2002,6(1):58-73P
[128]高平安,蔡自兴,余伶俐.一种基于多子群的动态优化算法[J].中南大学学报.2009,40(3)：731-736页
[129]李方方,赵英凯.基于小生境粒子群的多峰函数全局优化算法的研究[J].机械与电子.2007,1：58-60页
[130]于会,于忠,李伟华.基于小生境粒子群技术的多航迹规划研究[J].西北工业大学学报.2010,28(3)：415-420页
[131]许丽艳.基于空间收缩的粒子群优化算法及其在投资预测中的应用[D].哈尔滨工程大学硕士学位论文.2006：1-60页
[132]张磊,高尚.基于精英粒子群优化算法的图像分割方法[J].计算机应用与软件.2009,26(12)：89-92页
[133]高飞.基于空间收缩的种群灭亡差异演化算法[J].复杂系统与复杂性科学.2004,1(2)：87-92页
[134]宗志雄,高飞.基于空间收缩的种群灭亡精英演化算法[J].武汉化工学院学报.2005,27(1)：87-90页
[135]蔡荣英,李丽珊,林晓宇.求解旅行商问题的自学习粒子群优化算法[J].计算机工程与设计.2007,28(2)：261-263页
[136]王淑娟,沈继红,李焱.基于粒子群优化算法的二阶系统解耦[J].武汉理工大学学报.2010,34(2)：276-279页
[137]孟祥印,黄胜,李焱.混沌PSO分析及其在船舶设计中的应用[J].计算机工程与应用.2010,46(17)：224-228页
[138]李焱.基于函数变换的改进混沌粒子群优化[J].计算机应用研究.2010,27(11)：4105-4107页
[139]刘淳安.解多目标优化问题的新粒子群优化算法[J].计算机工程与应用.2006,42(2)：30-32页
[140]蒋浩,郑金华,陈良军.一种求解多目标优化问题的粒子群算法[J].模式识别与人工智能.2007,20(5)：606-611页
[141]蒋程涛,邵世煌.基于适配粒子群的多目标优化方法[J].计算机工程.2007,33(21)：1-2页
[142]Rabbani M. A multi-objective particle swarm optimization for project selection problem[J]. Expert Systems with applications, 2009,37(1):315-321P
[143]李丹,高立群,马佳,李扬.基于动态双种群粒子群算法的柔性工作车间高度[J].东北大学学报.2007,28(9)：1238-1242页
[144]loan C T. The particle swarm optimization algorithm:convergence analysis and parameter selection[J]. Information Processing Letters,2003,85 (6):317-325P
[145]陈炳瑞,冯夏庭.压缩搜索空间与速度范围粒子群优化算法[J].东北大学学报.2005,26(5)：488-491页
[146]庞淑萍.一种速度不完全更新混沌粒子群优化算法[J].计算机仿真.2010,9：215-219页
[147]Wang G Y.Variable velocity limit chaotic particle swarm optimizataion[C].2010 IEEE International conference on information and automation, ICIA2010, Univ of Harbin Engineering University,2010. Harbin,2010:1661-1666P
[148]宁必锋,褚国娟,马春丽,钱伟懿.一中改进的混合粒子群优化算法[J].渤海大学学报.2010,1：37-43页
[149]罗华飞著.Matlab GUI设计学习手记[M].第一版.北京：北京航空航天大学出版社,2009：1-142页
[150]刘道华,原思聪,张锦华,吴涛.粒子群参数自适应调整的优化设计[J].农业机械学报.2008,39(9)：134-137页
[151]李宁,邹彤,孙德宝.基于粒子群的多目标优化算法[J].计算机工程与应用.2005,41(23)：43-46页
[152]伟懿,李阿军,杨宁宁.基于混沌的多目标粒子群优化算法[J].计算机工程与设计.2008,29(18)：4794-4796页
[153]李中凯,谭建荣,冯毅雄.基于拥挤趴离排序的多目标粒f群优化算法及其应用[.J].计算机集成制造系统.2008,4(7)：1329-1336页
[2]Chou C M, Su K T, Tseng C H. Effectiveness of street sweeping and washing for controlling ambient TSP[J]. Atmospheric Environment, 2005,39(10):1891-1902P
[3]Engebretsen L, Karpinski M. TSP with bounded metrics[J]. Journal of Computer and System Sciences,2006,72(4):509-546P
[4]Wang J P, Hu M. A solution for TSP based on artificial fish algorithm[C]. Proceedings of the 2009 International Conference on Computational Intelligence and Natural Computing,2009. Wuhan,2009: 26-29P
[5]肖东喜,朱金福.飞机排班中航班环的动态构建方法[J].系统工程.2007,25(11)：19-25页
[6]刘家学,郑昌义,刘耀武.带有约束的运输问题及其推广应用[J].系统工程理论与实践.2002,22(2)：127-130页
[7]Krishnan P. Automatic synthesis of schedulers in timed systems[J]. Electronic Notes in Theoretical Computer Science,2000,31:118-131P
[8]叶建芳,王正肖,潘晓弘.免疫粒子群优化算法在车间作业调度中的应用[J].浙江大学学报.2008,42(5)：863-868页
[9]韩世莲,刘新旺.多目标多模式模糊运输问题的最优折衷解[J].系统工程.2007,25(9)：26-32页
[10]温强,胡明明,桑楠.基于彩色线阵CCD的大米色选算法[J].农业机械学报.2008,39(10)：105-108页
[11]陈孝敬,吴迪,何勇.基于小波包和偏最小二乘支持向量机的多光谱纹理图像的大米分类研究[J].光谱学与光潜分析.2009,29(1)：222-225页
[12]万仲平,费浦生编著.优化理论与方法[M].第一版.武汉：武汉大学出版 社,2004：10-62页
[13]李春明编著.优化方法[M].第一版.南京：东南大学出版社,2009：10-50页
[14]Eiben A E. Genetic algorithms with multi-parent recombination[C]. Proceedings of the International Conference on Evolutionary Computation,1994. Orlando,1994:78-87P
[15]Syswerda, G. Uniform crossover in genetic algorithms[C]. Proceedings of the Third International Conference on Genetic Algorithms, George mason University,1989. San Francisco:Morgan Kaufmann Publishers Inc,1989:2-9P
[16]Wu A, Tsang P W, Yuen T Y, Yeung L F. Affine invariant object shape matching using genetic algorithm with multi-parent orthogonal recombination and migrant principle[J]. Applied Soft Computing Journal,2009,9(1):282-289P
[17]元昌安,彭昱忠等编著.基因表达式编程原理与应用[M].第一版.北京：科学出版社,2010：1-170页
[18]玄光男,程润伟,于歆杰等编著.遗传算法与工程优化[M].第·版.北京：清华大学出版社,2004：1-52页
[19]姜大立,杨西龙,杜文.车辆路径问题的遗传算法研究[J].系统工程理论与实践.1999,19(6)：40-45页
[20]曾相戈,韩伯棠.一种求解带资源约束的并行机器多目标调度问题的遗传算法[J].系统工程理论与实践.2005,25(9)：78-82页
[21]韩丽霞,王宇平.解旅行商问题的一个新的遗传算法[J].系统工程理论与实践.2007,27(12)：145-150页
[22]胡大伟,陈诚.遗传算法(GA)和禁忌搜索算法(TS)在配送中心选址和路线问题中的应用[J].系统工程理论与实践.2007,27(9)：171-176页
[23]许项东,程琳.城市道路单行系统布局优化的双层规划模型和混合算法[J].系统工程理论与实践.2009,29(10)：180-187页
[24]江中央,蔡自兴,王勇.求解全局优化问题的混合自适应正交遗传算法[J].软件学报.2010,21(6)：1296-1307贞
[25]Farasat A, Menhaj M B, Mansouri T, Moghadam M R S. A new model-free optimization algorithm inspired from asexual reproduction[J]. Applied Soft Computing Journal,2010,10(4):1284-1292P
[26]Kirkpatrick S C D, Gelatt M P, Vecchi. Optimization by Simulated Annealing[J]. Science,1983,220(4598):671-680P
[27]Cerny V. A thermodynamical approach to the travelling salesman problem:an efficient simulation algorithm[J]. Journal of Optimization Theory and Applications,1985,45:41-51P
[28]Granville V, Krivanek J P, Rasson. Simulated annealing:A proof of convergence[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence,1994,16(6):652-656P
[29]Vicente J D, Lanchares J, Hermida R. Placement by Thermodynamic Simulated Annealing[J]. Physics Letters A,2003,317(5):415-423P
[30]王银年,葛洪伟.求解TSP问题的改进模拟退火遗传算法[J].计算机工程与应用.2010,46(5)：44-47页
[31]杨卫波,赵燕伟.求解TSP问题的改进模拟退火算法[J].计算机工程与应用.2010,46(15)：34-36页
[32]许小勇.混沌模拟退火算法在数值函数优化中的应用[J].计算机与数字工程.2010,38(3)：37-40页
[33]冯洪奎,鲍劲松,金烨.混合粒子群优化算法求解多车辆拖动货物问题[J].计算机集成制造系统.2010,16(7)：1427-1436页
[34]许智宏,宋勃,郭艳艳.模拟退火与蚁群混合并行算法解旅行商问题[J].河北工业大学学报.2010,39(2)：48-51页
[35]肖思和,鲁红英,范安东.模拟退火算法在求解组合优化问题中的应用研究[J].四川理工学院学报.2010,23(1)：1296-1307页
[36]Glover F. Tabu Search-Part Ⅰ[J]. ORSA Journal on Computing,1989, 1 (3):190-206P
[37]Glover F. Tabu Search-Part Ⅱ[J]. ORSA Journal on Computing,1990, 2(1):4-32P
[38]Cvijovic D, Klinowski J. Taboo search-an approach to the multiple minima problem[J]. Science,1995,267:664-666P
[39]董宗然,陈明华,李迎秋.最短路径问题的禁忌搜索求解方法[J].计算机工程与应用.2010,46(33)：36-38页
[40]李俊青,潘全科,王玉亭.多目标柔性车间调度的Pareto混合禁忌搜索算法[J].计算机集成制造系统.2010,16(7)：1419-1426页
[41]刘霞,齐欢.基于禁忌搜索的动态车辆路径问题研究[J].武汉理工大学学报.2010,34(2)：293-296页
[42]张芬,谢安世,周传华.适于高维空间搜索的自组织学习算法[J].计算机工程与设计.2010,31(9)：2005-2009页
[43]张汉强,卢建刚,陈金水.新的混合智能优化算法及其多目标优化应用[J].计算机应用.2010,30(5)：1290-1292页
[44]Zong W G, Joong H K, Loganathan G V. A New Heuristic Optimization Algorithm:Harmony Search[J]. Simulation,2001,76(2):60-68P
[45]李亮,迟世春,林皋.改进和声搜索算法及其在土坡稳定分析中的应用[J].土木工程学报.2006,39(5)：107-111页
[46]金永强,苏怀智,李子阳.基于和声搜索的边坡稳定性投影寻踪聚类分析[J].水利学报.2007,1：682-686页
[47]Marco D, Thomas S编著.蚁群优化[M].第一版.北京：清华大学出版社,2007：1-112页
[48]Dorigo M. Optimization, Learning and Natural Algorithms [D]. Italie Politecnico di Milano,1992:45-47P
[49]Dorigo M, Maniezzo V. Colorni, Ant system:optimization by a colony of cooperating agents[J]. IEEE Transactions on Systems, Man, and Cybernetics-Part B,1996,26(1):29-41P
[50]Dorigo M, Gamhardella L M. Ant Colony System:A Cooperative Learning Approach to the Traveling Salesman Problem[J]. IEEE Transactions on Evolutionary Computation,1997,1(1):53-66P
[51]Caro G D, Dorigo M. AntNet:a mobile agents approach to adaptive routing[C]. Proceedings of the Thirty-First Hawaii International Conference on System Science,1998:74-83P
[52]Bseten M, Stutzle T, Dorigo, M. Ant colony optimization for the total weighted tardiness prohlem[J]. Lecture Notes in Computer Science, 2000,1917:611-620P
[53]刘利强.蚁群优化方法研究及其在潜艇导航规划中的应用[D].哈尔滨工程 大学博士学位论文.2008：1-60页
[54]http://demonstrations.wolfram.com/FoodSearchingModelForAnts/
[55]Parpinelli R S, Lopes H S, Freitas A. Data mining with an ant colony optimization algorithm[J]. IEEE Transaction on Evolutionary Computation,2002,6(2):321-332P
[56]Martens D, Backer D M, Haesen R, Vanthienen J. Classification with Ant Colony Optimization[J]. IEEE Transactions on Evolutionary Computation,2007,11(5):651-665P
[57]Donati A V, Montemanni R, Casagrande N, Rizzoli A E, Gambardella L M. Time Dependent Vehicle Routing Problem with a Multi Ant Colony System[J]. European Journal of Operational Research,2008,185(3): 1174-1191P
[58]Chen P W, Zhang J. Ant Colony Optimization Approach to Grid Workflow Scheduling Problem with Various QoS Requirements[J]. IEEE Transactions on Systems, Man, and Cybernetics-Part C:Applications and Reviews,2009,31(1):29-43P
[59]Abbaspour K C, Schulin R, Van G M T. Estimating unsaturated soil hydraulic parameters using ant colony optimization[J]. Advances In Water Resources,2001,24(8):827-841P
[60]赵书强,王磊.改进蚁群在配电网规划中的应用[J].电力系统保护与控制.2010,24：61-65页
[61]李丽,牛奔等编著.粒子群优化算法[M].第一版.北京：冶金工业出版社,2009：1-70页
[62]Balas E, Vazacopoulos A. Guided local search with shifting bottleneck for job shop scheduling[J]. Management Science,1998, 44(2):262-274P
[63]Voudouris C, Tsang E. Guided local search with shifting bottleneck for job shop scheduling[J]. European Journal of Operational Research,1999,113(2):469-499P
[64]Flores L, Gilberto, Reed, Martin J. Guided local search as a network planning algorithm that incorporates uncertain traffic demands [J]. Computer Networks,2007,51(11):3172-3196P
[65]Hu G, Xiong W, Jiang B, Yuan J, Zhang X. Binary ant colony algorithm with balanced search bias[C]. Proceedings of the World Congress on Intelligent Control and Automation,2010:3120-3125P
[66]Lara A, Sanchez G, Coello C A C, Schutze 0. A new local search strategy for memetic multiobjective evolutionary algorithms [J]. IEEE Transactions on Evolutionary Computation,2010,14(1): 112-132P
[67]李兰英,张雷雷,石敏.改进的二维增强贪婪软硬件划分算法[J].计算机工程与应用.2009,45(21)：64-67页
[68]刘星宝,蔡自兴,王勇.用于全局优化问题的混合免疫进化算法[J].西安电子科技大学学报.2010,37(5)：971-980页
[69]燕乐纬,陈树辉,李森.一种改进的广义遗传算法及其在鲁棒优化问题中的应用[J].振动与冲击.2010,29(4)：30-33页
[70]王子若,陈永昌编著.优化计算方法[M].第一版.北京：机械工业出版社,1989：1-42页
[71]沈继红.基于光学原理的优化方法的研究[C].哈尔滨工程大学理学院学术年会.黑龙江哈尔滨,2007.哈尔滨：哈尔滨工程大学,2007：20-21页
[72]Shen J H, Li Y. Optimization Algorithm Based on Optical Principles[C].国际一般系统研究会中国分会,Harbin University of Science and Technology,2007. Harbin:Harbin University of Science and Technology Pr,2007:20-24P
[73]Shen J H, Li Y. An optimization algorithm based on optical princ.iples[J]. Advances in Systems Science and Applications,2009, 9(4):647-655P
[74]Shen J H, Li Y, Light Ray Optimization and its parameters analysis[C]. 2009 International Joint Conference on computational sciences and optimization, Sanya, Hainan.2009,2:918-922P
[75]沈继红,李焱.基于正六边形网格的光线寻优算法[C].中国运筹学会成立三十周年暨2010学术年会,2010,北京.2010：21-22页
[76]Shen J H, Li J L. The principle analysis of Light Ray Optimization Algorithm[C].2010 Second International Conference on Computational Intelligence and Natural Computing Proceedings. CINC, China,2010,2:154-157P
[77]Shen J H, Li Y. Light Ray Optimization with Function Transform[C]. The 8th International conference on Optimization: Techniques and Applications. Shanghai.2010:21-22P
[78]梁柱编著.光学原理教程[M].第一版.北京：北京航空航天大学出版社,2005：1-70页
[79]盛国.推动波动光学发展的几个重要实验[J].物理通报.2007,9(1)：50-52页
[80]单天明.从波动说和微粒说之争谈假说的科学功能[J].学理论.2011,4：43-44页
[81]Y. H. Li, Energics and Philosophy [M]. ShanXi Science and Technology Press. Taiyuan.2005:1-30P
[82]Richard S. Optical Refraction and Fermat's Principle at a Point [J]. Journal of the Optical Society of America.1950,40:244-245P
[83]塔拉.最小作用量原理与简单性原则[J].内蒙古大学学报.2003,35(1)：116-120页
[84]张慧鹏.物理学中的一个原理一最小作用量原理[J].物理通报.2009,5：16-18页
[85]Tan K, Liang C, Shi X. Optic eikonal Fermat's principle and the least action principle[J]. Science in China Series G:Physics, Mechanics and Astronomy.2008.51:1826-1834P
[86]Ville R. I Kaila, Arto Annila. Natural selection for least act ion [J]. Proceedings of the royal society A.2008.464:3055-3070P
[87]http://en.wikipedia.org/wiki/speed_of_light.
[88]吴佩萱.光传播中的极值问题[J].物理与工程.2005,15(5)：21-23页
[89]谢申监,周积义译.多维空间画法几何及其应用[M]北京：清华大学出版社,1983：1-10页
[90]Xu L Y, Shen J H, Bi X J. A novel particle swarm optimizer based on space contraction [J]. Journal of Harbin Engineering Unviersity, 2006,27:542-546P
[91]Winston P H. Artificial Intelligence[M] 3rd edition, Addison-Welley, Reading MA,1992:1-20P
[92]Ali T, Al A, Azzedine Z. A new modified particle swarm optimization algorithm for adaptive equal ization[J]. Digital Signal Processing, 2011,21(2):195-207P
[93]Ali A, Atai. On the limitations of classical benchmark functions for evaluating robustness of evolutionary algorithms[J]. Applied Mathematics and Computation,2010,215:3222-3229P
[94]Gaviano M, Lera D. Test. functions wi th variable attraction regions or global optimization problems[J]. Journal of global Optimization, 1998,13:207-223P
[95]Ahrari A, Shariat M, Atai A. GEM:a novel evolutionary optimization method with improved neighbourhood search[J]. Appl. Math. Comput, 2009,210:376-386P
[96]Yang G Y, Guang Y. Particle swarm optimization with mutation for best position of particles[J]. Journal of Harbin Engineering University,2006,27:531-536P
[97]Zhang H, Zhang T N, Shen J H. Research on decision-makings of structure optimization based on improved Tent PSO[J]. control and Decision,2008,23(8):857-862P
[98]Gao F, Cui G. Novel multi-step position-selectable updating particle swarm optimization algori thm[J]. Acta Electronica Sinica, 2009,37(3):529-534P
[99]Li H, He X, Xie X L. A new boundary condition for particle swarm optimization[J]. Journal of Convergence Information Technology, 2010,5(9):21-22P
[100]Ghorng Y S, Yi P H. Efficient hexagonal inner search for fast motion estimation[C]. International Conference on Image Processing,2005, Genova, Italy.2005:1093-1096P
[101]Quan E, Lalush D S. Evaluation of hexagonal and square geometries for motion-free arrayed-source X-ray micro-CT[C].2007 4th IEEE International Symposium on Biomedical Imaging:Macro to Nano,2007, Arlington, VA, USA.2007:221-224P
[102]梁森,陈花玲.常见蜂窝胞元轴向承载能力研究[J].四川兵工学报.2011,32(1)：65-69页
[103]Crypton. The mathematics of the honeycomb[J]. Science Digest.1985, 93 (6):74-77P
[104]Jie J, Zeng J C, Han C Z, Wang Q H. Knowledge-based cooperative particle swarm optimization[J]. Applied Mathematics and Computation,2008,205(2):861-873P
[105].Selleri S, Mussetta M, Pirinoli P, Zich R. Some insight over new variations of the particle swarm optimization method[J]. IEEEE Atennas and wireless propagationletters,2006,5(1):235-238P
[106]LiuXY, Liu H, Duan H C. Particle swarm optimization based on dynamic niche technology with applications to conceptual design[J]. Advances in Engineering Software,2007,38(10):668-676P
[107]Jiang M, Yang S Y. Stochastic convergence analysis and parameter selection of the standard particle swarm optimization algorithm[J]. Information Processing Letters,2007,102(1):8-16P
[108]Carlisle A, Doziert G. An off-the-shelf PSO[C]. Proceedings of the Workshop on Particle Swarm Optimization. Indianapolis, USA,2001: 1-6P
[109]曾建潮,介婧,崔志华.微粒群算法[M].北京：科学出版社,2004：10-11,104-120页
[110]张丽平,俞欢军,陈德钊,胡上序.粒子群优化算法的分析与改进[J].信息与控制.2004,33(5)：513-517页
[111]傅阳光,周成平,丁明跃.标准粒子群优化算法的收敛性分析[J].应用数学.2011,24(1)：187-194页
[112]Kennedy J, Eberhart R. Particle swarm optimization[C]. IEEE International Conference on Neural Networks-Conference Proceedings,1995:1942-1948P
[113]Kennedy J, Eberhart R. New optimizer using particle swarm theory[C]. Proceedings of the International Symposium on Micromechatronics and Human Science,1995:39-43P
[114]Shi Y, Eberhart R. Modified particle swarm optimizer[C]. Proceedings of the TEEE Conference on Evolutionary Computation, ICEC,1998:69-73P
[115]Reeves W T. Particle systems-a technique for modeling a class of fuzzy objects [J]. ACM Transactions on Graphics,1983,2(2):91-108P
[116]闵克学,葛宏伟,张毅,梁艳春.基于蚁群和粒子优化的混合算法求解TSP问题[J].吉林大学学报.2006,24(4)：402-405页
[117]柴宝杰,刘大为.基于粒子群优化的蚁群算法在TSP中的应用[J].计算机仿真.2009,26(8)：89-91页
[118]郑洁,李凯,李晓.用于求解对称旅行商问题的粒子群算法和蚂蚁算法的融合[J].计算机应用与软件.2010,27(1)：224-227页
[119]李宁,孙德宝,岑翼刚,邹彤.带变异算子的粒子群优化算法[J].计算机工程与应用.2004,(17)：12-14页
[120]张炯,刘天琪,苏鹏.基于遗传粒子群混合算法的机组组合优化[J].电力系统保护与控制.2009,37(9)：25-29页
[121]Shi Y H, Eberhart R C. Parameter selection in particle swarm optimization[C]. Proceedings of the Seventh Annual Conference on Evolutionary Programming. New York,1998:591-600P
[122]Shi Y H, Eberhart R C. Empirical study of particle swarm optimization[J]. Proceedings of the IEEE Congress on Evolutionary Computation. Piscataway, NJ:IEEE ServiceCenter,1999:1945-1950P
[123]Clerc M. The swarm and the queen:towards a deterministic and adaptive particle swarm optimization[C]. Proceedings of the IEEE Congress on Evolutionary Computation. IEEE Press,1999.1951-1957P
[124]杨维,李歧强.粒子群优化算法综述[J].中国工程科学.2004,5：87-94页
[125]纪震,周家锐,廖惠连等.智能单粒子优化算法[J].计算机学报.2010,33(3)：556-561页
[126]孙传峰,周刘喜.粒子群优化中最大速度选择[J].计算机仿真.2007,24(5)：162-164页
[127]Clerc M, Kennedy J. The particle swarm-explosion, stability, and convergence in a multidimensional complex space[J]. IEEE Transactions on Evolutionary Computation,2002,6(1):58-73P
[128]高平安,蔡自兴,余伶俐.一种基于多子群的动态优化算法[J].中南大学学报.2009,40(3)：731-736页
[129]李方方,赵英凯.基于小生境粒子群的多峰函数全局优化算法的研究[J].机械与电子.2007,1：58-60页
[130]于会,于忠,李伟华.基于小生境粒子群技术的多航迹规划研究[J].西北工业大学学报.2010,28(3)：415-420页
[131]许丽艳.基于空间收缩的粒子群优化算法及其在投资预测中的应用[D].哈尔滨工程大学硕士学位论文.2006：1-60页
[132]张磊,高尚.基于精英粒子群优化算法的图像分割方法[J].计算机应用与软件.2009,26(12)：89-92页
[133]高飞.基于空间收缩的种群灭亡差异演化算法[J].复杂系统与复杂性科学.2004,1(2)：87-92页
[134]宗志雄,高飞.基于空间收缩的种群灭亡精英演化算法[J].武汉化工学院学报.2005,27(1)：87-90页
[135]蔡荣英,李丽珊,林晓宇.求解旅行商问题的自学习粒子群优化算法[J].计算机工程与设计.2007,28(2)：261-263页
[136]王淑娟,沈继红,李焱.基于粒子群优化算法的二阶系统解耦[J].武汉理工大学学报.2010,34(2)：276-279页
[137]孟祥印,黄胜,李焱.混沌PSO分析及其在船舶设计中的应用[J].计算机工程与应用.2010,46(17)：224-228页
[138]李焱.基于函数变换的改进混沌粒子群优化[J].计算机应用研究.2010,27(11)：4105-4107页
[139]刘淳安.解多目标优化问题的新粒子群优化算法[J].计算机工程与应用.2006,42(2)：30-32页
[140]蒋浩,郑金华,陈良军.一种求解多目标优化问题的粒子群算法[J].模式识别与人工智能.2007,20(5)：606-611页
[141]蒋程涛,邵世煌.基于适配粒子群的多目标优化方法[J].计算机工程.2007,33(21)：1-2页
[142]Rabbani M. A multi-objective particle swarm optimization for project selection problem[J]. Expert Systems with applications, 2009,37(1):315-321P
[143]李丹,高立群,马佳,李扬.基于动态双种群粒子群算法的柔性工作车间高度[J].东北大学学报.2007,28(9)：1238-1242页
[144]loan C T. The particle swarm optimization algorithm:convergence analysis and parameter selection[J]. Information Processing Letters,2003,85 (6):317-325P
[145]陈炳瑞,冯夏庭.压缩搜索空间与速度范围粒子群优化算法[J].东北大学学报.2005,26(5)：488-491页
[146]庞淑萍.一种速度不完全更新混沌粒子群优化算法[J].计算机仿真.2010,9：215-219页
[147]Wang G Y.Variable velocity limit chaotic particle swarm optimizataion[C].2010 IEEE International conference on information and automation, ICIA2010, Univ of Harbin Engineering University,2010. Harbin,2010:1661-1666P
[148]宁必锋,褚国娟,马春丽,钱伟懿.一中改进的混合粒子群优化算法[J].渤海大学学报.2010,1：37-43页
[149]罗华飞著.Matlab GUI设计学习手记[M].第一版.北京：北京航空航天大学出版社,2009：1-142页
[150]刘道华,原思聪,张锦华,吴涛.粒子群参数自适应调整的优化设计[J].农业机械学报.2008,39(9)：134-137页
[151]李宁,邹彤,孙德宝.基于粒子群的多目标优化算法[J].计算机工程与应用.2005,41(23)：43-46页
[152]伟懿,李阿军,杨宁宁.基于混沌的多目标粒子群优化算法[J].计算机工程与设计.2008,29(18)：4794-4796页
[153]李中凯,谭建荣,冯毅雄.基于拥挤趴离排序的多目标粒f群优化算法及其应用[.J].计算机集成制造系统.2008,4(7)：1329-1336页
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