光线寻优算法的研究及改进
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摘要
优化问题大量存在于科学研究和工程应用中的各个领域，优化对象在复杂化和规模化等方面不断提高。以生物智能或自然现象为基础的智能算法因其具有简单通用、鲁棒性好、适于并行处理等特点，因此成为解决大规模复杂优化问题的有力工具。
基于费马原理，一种新型的智能优化算法——光线寻优算法被提出，它通过模拟光在变折射率介质中的传播过程进行寻优，为智能计算用于解决最优化问题提供了新思路。算法主要针对全局最优值不易搜索的难题，具有不涉及随机因素、可调参数少、结构简单、容易实现等优点。光线寻优算法首先用矩形网格划分搜索区域，并将每一网格填充进具有不同折射率的介质，即光线传播速度取为网格中心点所对应的目标函数值；然后将搜索路径设想为光的传播路径，并且认为光仅在各网格的边界上发生折射和反射，在各网格内部沿直线传播。当折射和反射同时发生时，取折射路径为寻优路径，仅当只发生反射时(即满足全反射条件)，取反射路径为寻优路径，按照此规则算法在搜索区域内自动搜索寻优。
优化方法的理论研究对完善算法体系、改进算法性能、拓宽算法应用领域具有重要作用，为此，本论文基于变分原理，对光线寻优算法的寻优机理、收敛性、稳定性进行分析，并将算法成功应用于求解函数优化问题。具体研究内容如下：
第一，对光线寻优算法的寻优机理进行分析，证明了光线会在水平和竖直分界线上交替进行折射，即设置矩形网格是有意义的；折射在算法中的作用是加速函数的变小趋势和减缓函数的变大趋势；搜索不会沿着函数值变大的方向一直进行下去，而会通过反射改变搜索方向。从理论角度上验证了算法的可行性。
第二，根据费马原理、利用变分法推导出基于折射的光线寻优算法最优化的实现过程，即对分层介质中算法的寻优功能做了具体的分析，得出了光具有偏向折射率增大方向、偏离折射率减小方向的自动寻优性质。从连续介质中的真实光线路径出发，通过分析光线所满足的方程，得真实光线同样具有寻优的功能，并分析光线寻优算法与光线方程所确定的寻优路径之间的关系，进而得出分块介质中算法亦具有自动寻优功能。
第三，对光线方程欧拉数值解法与光线寻优算法迭代公式的关系进行了研究，进而在光线寻优算法迭代公式中加入一项改进算法，这不仅使得精度提高一阶，而且加快了收敛的速度，解决了光线寻优算法推广到高维收敛速度变慢的问题。
第四，针对光线寻优算法局部搜索能力弱和收敛性理论完善困难的问题，提出了贪婪光线寻优算法，并通过理论推导证明该算法的局部收敛性。贪婪光线寻优与光线寻优算法的区别在于值变大即取反射路径为寻优路径，不接受“坏解”，适合求单极值问题。
第五，理论分析和数值实验表明：网格越小，求解精度越高，但相应的迭代次数增多，收敛速度变慢。针对这一问题改进算法，提出了基于变网格的光线寻优算法，算法搜索初期用较大的网格，确定全局最优点的大概位置后，换用较小的网格继续搜索。根据具体的精度要求，可以选择多次变小网格，从而达到提高收敛精度和速度的目的。
第六，将光线寻优算法用于求解2维、1维以及3维以上优化问题，与遗传算法、模拟退火算法、粒子群算法进行数值实验对比分析，并通过变分法分析了光线寻优算法用于求解1维优化问题时的局部收敛性。
第七，将模拟退火算法中的退火策略引入到光线寻优算法中而提出的一种新型混合优化算法—基于退火策略的光线寻优算法。通过引入无网格思想，即在算法寻优过程中无需生成网格，而是按照一定的规则直接确定折射或反射界面，提出了无网格光线寻优算法。并分别对这2种改进算法进行了数值实验的研究和对比分析。
There exist many optimization problems in various fields of scientific researchengineering application. Complication, scale and other aspects of optimization objects areincreasing. Intelligence algorithms, based on biological intelligence or natural phenomena,characterize in the simple and general usage, sounding robustness and fitting for parallelprocessing. Therefore, they become a powerful tool for solving large-scale complexoptimization problems.
Light ray optimization algorithm is a new intelligent optimization algorithm based onFermat’s principle. The algorithm searches the optimal solution by simulating the propagationprocess of light in gradient-index media, and provides a new idea for intelligent computingsolving optimization problems. The algorithm was proposed to solve the problem that globaloptimum is not easy to search. It has the advantages of not involving random factors, fewtuning parameters, simple structure, and being easy to realize. In light ray optimizationalgorithm, the searching area is divided by rectangular grids which are full of media withdifferent refractivities, that is let the propagation velocity of light in each grid be the objectivefunction value of some point in this grid. Then the searching path is assumed to be thepropagation path of light rays. Refraction and reflection merely occur on the boundaries ofgrids, and light rays propagate along straight lines in the same grid. The refraction path ischosen as optimal path when refraction and reflection occur simultaneously. The reflectionpath is chosen as optimal path when only reflection occurs, that is the conditions of totalreflection are satisfied. According to these rules, light rays propagate in these media to searchthe optimal value automatically in the algorithm.
Theoretical study of optimization methods is important to perfect algorithm systems,improve algorithm performance and widen its application fields. Therefore, optimizationmechanism, convergence and stability of light ray optimization algorithm were theoreticallyanalyzed based on variational principle. And then the algorithm was used for solving functionoptimization problems. The concrete research contents are as follows:
Firstly, optimization mechanism of light ray optimization algorithm was analyzed, andthe following conclusions were proved. Refractions in the algorithm alternatively occur inboth horizontal and vertical directions, that is the setting of rectangular grids is meanful. The decrease of objective function is accelerated and the increase of function is reduced byrefraction in algorithm. Reflection will inevitably occur if the search along the directions thatmake the increase of function value keep going. Feasibility of the algorithm was proved fromthe theoretic aspect.
Secondly, according to Fermat’s principle and variational method, realization ofminimization process was derived in light ray optimization algorithm based on refraction, thatis concrete analysis of optimization function of the algorithm was made in layered medium.Light rays have the auto optimization property that will get closer to the direction that makesrefractivity increase, and get further from the one that makes it decrease. From the real lightpath in continuous media and by analyzing the equations that light rays are satisfied, the papergot the conclusion that real light rays also have the optimization function. The relationbetween the searching path determined by light ray optimization algorithm and the onedetermined by light rays equation was analyzed, and auto optimization function of algorithmin partitioned medium was also derived.
Thirdly, the relation between Euler method of ray equations and the iterative formula oflight ray optimization was studied. Light ray optimization algorithm was improved by addinga term to the iterative formula of this algorithm, by which the accuracy is increased and orderand the convergence speed is accelerated. This solve the problem that light ray optimizationalgorithm has a slow convergence speed when it is used for computing high dimensionalproblems.
Fourthly, light ray optimization algorithm is an intelligent algorithm with the weak localoptimization ability and the difficulty of perfection of convergence theory. To solve theseproblems, greedy light ray optimization algorithm was proposed. Local convergence of theproposed algorithm was proved via theoretical derivation. Comparing with light rayoptimization algorithm, the only difference of greedy light ray optimization algorithm is thatreflection path is chosen as optimal path when function value increases. It doesn’t accept any“bad solution”, and is suitable for solving single extremum optimization problems.
Fifthly, theoretical analysis and numerical experiments show that the smaller the grid,the higher accuracy solution is obtained, but with the increasing iterative times and slowerconvergence speed. An improved algorithm named light ray optimization algorithm based onvariable grid was proposed to solve this problem. Large grids are used to determine the approximate position of global optimal point at the early stages of the optimization, and thensmaller grids are used to continue searching. Grid size can be reduced for many timesaccording to concrete precision requirement, which improves the convergence precision andspeed.
Sixthly, light ray optimization algorithm was used for solving two dimensional, onedimensional, and more than three dimensional optimization problems, and was comparedexperimentally with genetic algorithm, simulated annealing algorithm and particle swarmoptimization algorithm. Local convergence of light ray optimization algorithm solving onedimensional optimization problems was analyzed by variational method.
Seventhly, a new hybrid algorithm named light ray optimization algorithm based onannealing strategy was proposed by introducing the annealing strategy of simulated annealingalgorithm. By introducing grid free thought, light ray optimization based on grid free methodwas proposed. This algorithm doesn’t need grid generation, but determines refraction orreflection interfaces according to certain rules. Study and comparative analysis of numericalexperiments of these two improved algorithms were made respectively.
基于费马原理，一种新型的智能优化算法——光线寻优算法被提出，它通过模拟光在变折射率介质中的传播过程进行寻优，为智能计算用于解决最优化问题提供了新思路。算法主要针对全局最优值不易搜索的难题，具有不涉及随机因素、可调参数少、结构简单、容易实现等优点。光线寻优算法首先用矩形网格划分搜索区域，并将每一网格填充进具有不同折射率的介质，即光线传播速度取为网格中心点所对应的目标函数值；然后将搜索路径设想为光的传播路径，并且认为光仅在各网格的边界上发生折射和反射，在各网格内部沿直线传播。当折射和反射同时发生时，取折射路径为寻优路径，仅当只发生反射时(即满足全反射条件)，取反射路径为寻优路径，按照此规则算法在搜索区域内自动搜索寻优。
优化方法的理论研究对完善算法体系、改进算法性能、拓宽算法应用领域具有重要作用，为此，本论文基于变分原理，对光线寻优算法的寻优机理、收敛性、稳定性进行分析，并将算法成功应用于求解函数优化问题。具体研究内容如下：
第一，对光线寻优算法的寻优机理进行分析，证明了光线会在水平和竖直分界线上交替进行折射，即设置矩形网格是有意义的；折射在算法中的作用是加速函数的变小趋势和减缓函数的变大趋势；搜索不会沿着函数值变大的方向一直进行下去，而会通过反射改变搜索方向。从理论角度上验证了算法的可行性。
第二，根据费马原理、利用变分法推导出基于折射的光线寻优算法最优化的实现过程，即对分层介质中算法的寻优功能做了具体的分析，得出了光具有偏向折射率增大方向、偏离折射率减小方向的自动寻优性质。从连续介质中的真实光线路径出发，通过分析光线所满足的方程，得真实光线同样具有寻优的功能，并分析光线寻优算法与光线方程所确定的寻优路径之间的关系，进而得出分块介质中算法亦具有自动寻优功能。
第三，对光线方程欧拉数值解法与光线寻优算法迭代公式的关系进行了研究，进而在光线寻优算法迭代公式中加入一项改进算法，这不仅使得精度提高一阶，而且加快了收敛的速度，解决了光线寻优算法推广到高维收敛速度变慢的问题。
第四，针对光线寻优算法局部搜索能力弱和收敛性理论完善困难的问题，提出了贪婪光线寻优算法，并通过理论推导证明该算法的局部收敛性。贪婪光线寻优与光线寻优算法的区别在于值变大即取反射路径为寻优路径，不接受“坏解”，适合求单极值问题。
第五，理论分析和数值实验表明：网格越小，求解精度越高，但相应的迭代次数增多，收敛速度变慢。针对这一问题改进算法，提出了基于变网格的光线寻优算法，算法搜索初期用较大的网格，确定全局最优点的大概位置后，换用较小的网格继续搜索。根据具体的精度要求，可以选择多次变小网格，从而达到提高收敛精度和速度的目的。
第六，将光线寻优算法用于求解2维、1维以及3维以上优化问题，与遗传算法、模拟退火算法、粒子群算法进行数值实验对比分析，并通过变分法分析了光线寻优算法用于求解1维优化问题时的局部收敛性。
第七，将模拟退火算法中的退火策略引入到光线寻优算法中而提出的一种新型混合优化算法—基于退火策略的光线寻优算法。通过引入无网格思想，即在算法寻优过程中无需生成网格，而是按照一定的规则直接确定折射或反射界面，提出了无网格光线寻优算法。并分别对这2种改进算法进行了数值实验的研究和对比分析。
There exist many optimization problems in various fields of scientific researchengineering application. Complication, scale and other aspects of optimization objects areincreasing. Intelligence algorithms, based on biological intelligence or natural phenomena,characterize in the simple and general usage, sounding robustness and fitting for parallelprocessing. Therefore, they become a powerful tool for solving large-scale complexoptimization problems.
Light ray optimization algorithm is a new intelligent optimization algorithm based onFermat’s principle. The algorithm searches the optimal solution by simulating the propagationprocess of light in gradient-index media, and provides a new idea for intelligent computingsolving optimization problems. The algorithm was proposed to solve the problem that globaloptimum is not easy to search. It has the advantages of not involving random factors, fewtuning parameters, simple structure, and being easy to realize. In light ray optimizationalgorithm, the searching area is divided by rectangular grids which are full of media withdifferent refractivities, that is let the propagation velocity of light in each grid be the objectivefunction value of some point in this grid. Then the searching path is assumed to be thepropagation path of light rays. Refraction and reflection merely occur on the boundaries ofgrids, and light rays propagate along straight lines in the same grid. The refraction path ischosen as optimal path when refraction and reflection occur simultaneously. The reflectionpath is chosen as optimal path when only reflection occurs, that is the conditions of totalreflection are satisfied. According to these rules, light rays propagate in these media to searchthe optimal value automatically in the algorithm.
Theoretical study of optimization methods is important to perfect algorithm systems,improve algorithm performance and widen its application fields. Therefore, optimizationmechanism, convergence and stability of light ray optimization algorithm were theoreticallyanalyzed based on variational principle. And then the algorithm was used for solving functionoptimization problems. The concrete research contents are as follows:
Firstly, optimization mechanism of light ray optimization algorithm was analyzed, andthe following conclusions were proved. Refractions in the algorithm alternatively occur inboth horizontal and vertical directions, that is the setting of rectangular grids is meanful. The decrease of objective function is accelerated and the increase of function is reduced byrefraction in algorithm. Reflection will inevitably occur if the search along the directions thatmake the increase of function value keep going. Feasibility of the algorithm was proved fromthe theoretic aspect.
Secondly, according to Fermat’s principle and variational method, realization ofminimization process was derived in light ray optimization algorithm based on refraction, thatis concrete analysis of optimization function of the algorithm was made in layered medium.Light rays have the auto optimization property that will get closer to the direction that makesrefractivity increase, and get further from the one that makes it decrease. From the real lightpath in continuous media and by analyzing the equations that light rays are satisfied, the papergot the conclusion that real light rays also have the optimization function. The relationbetween the searching path determined by light ray optimization algorithm and the onedetermined by light rays equation was analyzed, and auto optimization function of algorithmin partitioned medium was also derived.
Thirdly, the relation between Euler method of ray equations and the iterative formula oflight ray optimization was studied. Light ray optimization algorithm was improved by addinga term to the iterative formula of this algorithm, by which the accuracy is increased and orderand the convergence speed is accelerated. This solve the problem that light ray optimizationalgorithm has a slow convergence speed when it is used for computing high dimensionalproblems.
Fourthly, light ray optimization algorithm is an intelligent algorithm with the weak localoptimization ability and the difficulty of perfection of convergence theory. To solve theseproblems, greedy light ray optimization algorithm was proposed. Local convergence of theproposed algorithm was proved via theoretical derivation. Comparing with light rayoptimization algorithm, the only difference of greedy light ray optimization algorithm is thatreflection path is chosen as optimal path when function value increases. It doesn’t accept any“bad solution”, and is suitable for solving single extremum optimization problems.
Fifthly, theoretical analysis and numerical experiments show that the smaller the grid,the higher accuracy solution is obtained, but with the increasing iterative times and slowerconvergence speed. An improved algorithm named light ray optimization algorithm based onvariable grid was proposed to solve this problem. Large grids are used to determine the approximate position of global optimal point at the early stages of the optimization, and thensmaller grids are used to continue searching. Grid size can be reduced for many timesaccording to concrete precision requirement, which improves the convergence precision andspeed.
Sixthly, light ray optimization algorithm was used for solving two dimensional, onedimensional, and more than three dimensional optimization problems, and was comparedexperimentally with genetic algorithm, simulated annealing algorithm and particle swarmoptimization algorithm. Local convergence of light ray optimization algorithm solving onedimensional optimization problems was analyzed by variational method.
Seventhly, a new hybrid algorithm named light ray optimization algorithm based onannealing strategy was proposed by introducing the annealing strategy of simulated annealingalgorithm. By introducing grid free thought, light ray optimization based on grid free methodwas proposed. This algorithm doesn’t need grid generation, but determines refraction orreflection interfaces according to certain rules. Study and comparative analysis of numericalexperiments of these two improved algorithms were made respectively.
引文
[1]何坚勇.最优化方法.北京:清华大学出版社,2007:313-346页
[2]陈宝林.最优化理论与算法.北京:清华大学出版社,2005:1-4页
[3] Abbasbandy S, Jafarian A. Steepest descent method for solving fuzzy nonlinearequations. Applied Mathematics and Computation.2006,174(1):669-675P
[4] Dai Z F. Two modified HS type conjugate gradient methods for unconstrainedoptimization problems. Nonlinear Analysis: Theory, Method&Applications.2011,74:927-936P
[5]刘小群.优化问题的智能算法及其哲学内涵.科学技术与辩证法.2007,24(6):92-102页
[6]王凌.智能优化算法及其应用.北京:清华大学出版社&施普林格出版社,2000:1-12页
[7]刑文训,谢金星.现代优化计算方法.北京:清华大学出版社,2005:13-27页
[8] Paszkowicz W. Properties of a genetic algorithm equipped with a dynamic penaltyfunction. Computational Materials Science.2009,45:77-83P
[9] Liang Y, Leung K S. Genetic algorithm with adaptive elitist-population strategies formultimodal function optimization. Applied Soft Computing.2011,11(2):2017-2034P
[10]刘道华,原思聪,邬长安,等.面向对象的改进遗传算法优化设计.华中科技大学学报(自然科学版).2008,36(7):89-92页
[11] Cai W W, Ewing D J, Ma L. Investigation of temperature parallel simulated annealingfor optimizing continuous functions with application to hyperspectral tomography.Applied Mathematics and Computation.2011,217(12):5754-5767P
[12] Saraiva J T, Pereira M L, Mendes V T, et al. A simulated annealing based approach tosolve the generator maintenance scheduling problem. Electric Power SystemsResearch.2011,81:1283-1291P
[13] Leung S C H, Zhang D F, Zhou C L, et al. A hybrid simulated annealing metaheuristicalgorithm for the two-dimensional knapsack packing problem. Computers&Operations Resarch.2012,39:64-73P
[14]印峰,王耀南,刘炜,等.个体速度差异的蚁群算法设计及仿真.智能系统学报.2009,4(6):528-533页
[15] Puris A, Bello R, Herrera F. Analysis of the efficacy of a Two-Stage methodology forant colony optimization: Case of study with TSP and QAP. Expert Systems withApplications.2010,37:5443-5453P
[16] Xiao J, Li L P. A hybrid ant colony optimization for continuous domains. ExpertSystems with Applications.2011:38(9):11072-11077P
[17] Chen C C. Two-layer particle swarm optimization for unconstrained optimizationproblems. Applied Soft Computing.2011,11:295-304P
[18] Thangaraj R, Pant M, Abraham A, et al. Particle swarm optimization: hybridizationperspectives and experimental illustrations. Applied Mathematics and Computation.2011,217:5208-5226P
[19]傅阳光,周成平,丁明跃.标准粒子群优化算的收敛性分析.应用数学.2011,24(1):187-194P
[20]沈继红.基于光学原理的优化方法的研究.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
[21] Dantzig G B. Linear programming under uncertainty. Management Science.1955,1(3):197-206P
[22] Dantzig G B, Orden G B A, Wolfe P. Note on linear programs. Pacific J Math.1955,1(5):183-195P
[23]陈军斌,杨悦.最优化方法.北京:中国石化出版社,2011:15-37页
[24]汪定伟,王俊伟,王洪峰,等.智能优化方法.北京:高等教育出版社,2007:2-7页
[25] Jiang M, Luo Y P, Yang S Y. Stochastic convergence analysis and parameter selectionof the standard particle swarm optimization algorithm. Information Processing Letters.2007,102:8-16P
[26] Liang Y C, Zhou C G, Wang Z S. An equivalent genetic algorithm based on extendedstrings and its convergence analysis. Information Sciences.2001,138:119-135P
[27] Chakraborty P, Das S, Roy G G, et al. On convergence of the multi-objective particleswarm optimizers. Information Sciences.2011,181:1411-1425P
[28] Schmitt L M. Theory of Genetic Algorithms II: models for genetic operators over thestring-tensor representation of populations and convergence to global optima forarbitrary fitness function under scaling. Theoretical Computer Science.2004,310:181-231P
[29] Xu B L, Zhu J H, Xu H G. An ant stochastic decision based particle filter and itsconvergence. Signal Processing.2010,90:2731-2748P
[30] Meise C. On the convergence of parallel simulated annealing. Stochastic Processesand their Applications.1998,76:99-115P
[31]张慧斌,王鸿斌,胡志军.基于定常线性迭代法的PSO算法收敛性分析.计算机工程与应用.2011,47(31):35-37页
[32]沈继红,王侃,胡波.粒子群优化算法的数学基础.武汉大学学报(理学版).2011,57(4):360-364页
[33]刘星,张雪峰,蔡勇,等.改进遗传算法求解焊位群完全遍历.机械与电子.2011(3):24-26页
[34]李军华,黎明.噪声环境下遗传算法的收敛性和收敛速度估计.电子学报.2011,39(8):1898-1902页
[35]申元霞,王国胤,曾传华.相关性粒子群优化模型.软件学报.2011,22(4):695-708页
[36]高尚,杨静宇.最短路的蚁群算法收敛性分析.科学技术与工程.2006,6(3):273-277页
[37] Wang L, Tang D B. An improved adaptive genetic algorithm based on hormonemodulation mechanism for job-shop scheduling problem. Expert Systems withApplications.2011,38:7243-7250P
[38] Wu P F, Gao L Q, Zou D X, et al. An improved particle swarm optimization algorithmfor reliability problems. ISA Transactions.2011,50:71-81P
[39] Wang Z J. Improved ant colony algorithm and its simulation study. Energy Procedia.2011,11:4587-4592P
[40] Giovanni L D, Pezzella F. An improved genetic algorithm for the distributed andflexible job-shop scheduling problem. European Journal of Operational Research.2010,200:395-408P
[41] Han Y F, Shi P F. An improved ant colony algorithm for fuzzy clustering in imagesegmentation. Neurocomputing.2007,70:665-671P
[42] Gao W F, Liu S Y. Improved artificial bee colony algorithm for global optimization.Information Processing Letters.2011,111:871-882P
[43] Zhang K, Xu J, Geng X T, et al. Improved taboo search algorithm for designing DNAsequences. Progress in Natural Science.2008,18:623-627P
[44] Jiang Y, Hu T S, Huang C C, et al. An improved particle swarm optimizationalgorithm. Applied Mathematics and Computation.2007,193:231-239P
[45] Shintemirov A, Tang W J, Tang W H, et al. Improved modelling of power transformerwinding using bacterial swarming algorithm and frequency response analysis. ElectricPower Systems Research.2010,80:1111-1120P
[46] Yang J G, Zhuang Y B. An improved ant colony optimization algorithm for solving acomplex combinatorial optimization problem. Applied Soft Computing.2010,10:653-660P
[47] Wang Y, Zhou J Z, Qin H, et al. Improved chaotic particle swarm optimizationalgorithm for dynamic economic dispatch problem with valve-point effects. EnergyConversion and Management.2010,51:2893-2900P
[48] Shelokar P S, Siarry P, Jayaraman V K, et al. Particle swarm and ant colonyalgorithms hybridized for improved continuous optimization. Applied Mathematicsand Computation.2007,188:129-142P
[49] Katagiri H, Hayashida T, Nishizaki I, et al. A hybrid algorithm based on tabu searchand ant colony optimization for k-minimum spanning tree problems. Expert Systemswith Applications.2012,39:5681-5686P
[50] Kim D H, Abraham A, Cho J H. A hybrid genetic algorithm and bacterial foragingapproach for global optimization. Infromation Sciences.2007,177:3918-3937P
[51] Mhamdi B, Grayaa K, Aguili T. Hybrid of genetic algorithm with particle swarmoptimization to shape reconstruction of perfect conducting cylinders. InternationalJournal of Electronics and Communications.2011,65:1032-1039P
[52] Kao Y T, Zahara E. A hybrid genetic algorithm and particle swarm optimization formultimodal functions. Applied Soft Computing.2008,8:849-857P
[53] Sun J P, Fang L, Han J. Optimization of concrete hollow brick using hybrid geneticalgorithm combining with artificial neural networks. International Journal of Heat andMass Transfer.2010,53:5509-5518P
[54] Kiran M S, zceylan E, Gündüz M, et al. A novel hybrid approach based on particleswarm optimization and ant colony algorithm to forecast energy demand of Turkey.Energy Conversion and Management.2012,53:75-83P
[55] Marinakis Y, Marinaki M. A hybrid genetic-particle swarm optimization algorithm forthe vehicle routing problem. Expert Systems with Applications.2010,37:1446-1455P
[56] Rasmussen T K, Krink T. Improved Hidden Markov Model training for multiplesequence alignment by a particle swarm optimization-evolutionary algorithm hybrid.BioSystems.2003,72:5-17P
[57] Lee K S, Geem Z W. A new meta-heuristic algorithm for continuous engineeringoptimization: harmony search theory and practice. Computer methods in appliedmechanics and engineering.2005,194:3902-3933P
[58] Fotakis D, Sidiropoulos E. A new multi-objective self-organizing optimizationalgorithm (MOSOA) for spatial optimization problems. Applied Mathematics andComputation.2012,218:5168-5180P
[59] Farasat A, Menhaj M B, Mansouri T, et al. ARO: A new model-free optimizationalgorithm inspired from asexual reproduction. Applied Soft Computing.2010,10:1284-1292P
[60] Yildiz A R. A new design optimization framework based on immune algorithm andTaguchi’s method. Computers in Industry.2009,60:613-620P
[61] Pan W T. A new fruit fly optimization algorithm: taking the financial distress model asan example. Knowledge-Based Systems.2012,26:69-74P
[62] Lebaal N, Nouari M, Ginting A. A new optimization approach based on Kriginginterpolation and sequential quadratic programming algorithm for end millingrefractory titanium alloys. Applied Soft Computing.2011,11:5110-5119P
[63] Ochoa S, Wozny G, Repke J U. A new algorithm for global optimization:Molecular-Inspired Parallel Tempering. Computers and Chemical Engineering.2010,34:2072-2084P
[64] Oftadeh R, Mahjoob M J, Shariatpanahi M. A novel meta-heuristic optimizationalgorithm inspired by group hunting of animals: Hunting search. Computers andMathematics with Applications.2010,60:2087-2098P
[65] Holland J H. Adaptation in natural and artificial systems. Michigan: University ofMichigan Press,1975
[66] Whitley D. A genetic algorithm tutorial. Statistics and Computing.1994,4:65-85P
[67] Park C H, Lee W I, Han W S, et al. Improved genetic algorithm for multidisciplinaryoptimization of composite laminates. Computers and Structures.2008,86:1894-1903P
[68] Aryanezhad M B, Hemati M. A new genetic algorithm for solving nonconvexnonlinear programming problems. Applied Mathematics and Computation.2008,199(1):186-194P
[69] Li F C, Xu L D, Jin C X, et al. Structure of Multi-Stage Composite Genetic Algorithm(MSC-GA) and its performance. Expert Systems with Applications.2011,38(7):8929-8937P
[70] Kirkpatrick S, Gelatt C D, Vecchi M P. Optimization by simulated annealing. Science.1983,220(4598):671-680P
[71] Fabian V. Simulated annealing simulated. Computers and Mathematics withApplications.1997,33:81-94P
[72] Li N, Cha J Z, Lu Y Q. A parallel simulated annealing algorithm based on functionalfeature tree modeling for3D engineering layout design. Applied Soft Computing.2010,10:592-601P
[73] Hong W C. Traffic flow forecasting by seasonal SVR with chaotic simulatedannealing algorithm. Neurocomputing.2011,74:2096-2107P
[74]朱颢东,钟勇.一种改进的模拟退火算法.计算机技术与发展.2009,19(6):32-35页
[75] Glover F. Future paths for integer programming and links to artificial intelligence.Computers and Operations Research.1986,13(5):533-549P
[76]张晓菲,张火明.基于连续函数优化的禁忌搜索算法.中国计量学院学报.2010,21(3):251-256页
[77] Fescioglu-Unver N, Kokar M M. Self controlling tabu search algorithm for thequadratic assignment problem. Computers and Industrial Engineering.2011,60(2):310-319P
[78] Dorigo M. Optimization, learning and natural algorithms. PhD thesis, Politecnico diMilano, Italie,1992
[79] Colorni A, Dorigo M, Maniezzo V. Distributed optimization by ant colonies.Proceedings of the First European Conference on Artificial Life. Paris, France:Elsevier,1992:134-142P
[80] Colorni A, Dorigo M, Maniezzo V. An investigation of some properties of an antalgorithm. Proceedings of the Parallel Problem Solving from Nature Conference.Brussels, Belgium: Elsevier Publishing,1992:509-520P
[81]姜秋霞,王中杰.混合蚁群算法的研究及其应用.装备制造技术.2008(2):36-38页
[82] Neumann F, Witt C. Ant colony optimization and the minimum spanning tree problem.Theoretical Computer Science.2010,411(25):2406-2413P
[83] Ebergart R C, Kennenedy J. Particle swarm optimization. Proceedings of IEEEInternational Conference on Neural Networks. Perth, Australia,1995:1942-1948P
[84] Eberhart R, Kennedy J. A new optimizer using particle swarm theory. SixthInternational Symposium on Micro Machine and Human Science,1995:39-45P
[85]姜海明,谢康,王亚非.按概率突跳的改进微粒群优化算法.吉林大学学报(工学版).2007,37(1):141-145页
[86]田明,刘国志.惯性权重线性调整的局部收缩微粒群算法.吉林大学学报(理学版).2008,46(1):96-100页
[87]李剑,王乘.一种改进的自适应微粒群优化算法.华中科技大学学报(自然科学版).2008,36(3):118-121P
[88] Chen M R, Li X, Zhang X, et al. A novel particle swarm optimizer hybridized withextremal optimization. Applied Soft Computing.2010,10:367-373P
[89]秦洪德,石丽丽.一种新型的被动启发式粒子群优化算法.哈尔滨工程大学学报.2010,31(10):1298-1302页
[90]刘宏达,马忠丽.均匀粒子群算法.智能系统学报.2010,5(4):336-341P
[91] Linhares A. State-space search strategies gleaned from animal behavior: a travelingsalesman experiment. Biological Cybernetic.1998,78(3):167-173P
[92] Linhares A. Preying on optima: a predatory search strategy for combinatorial problem.Proc of IEEE International Conference on Systems, Man and Cybemetics. Piscataway,NJ: IEEE Press,1998:2974-2978P
[93] Linhares A. Synthesizing a predatory search strategy for VLSI layouts. IEEE Trans onEvolutionary Computation.1999,3(2):147-152P
[94]姚世伟,陈贤,沈海鸿.带变异特征的捕食搜索算法及其在TSP问题实验研究.科学技术与工程.2011,11(27):6652-6655P
[95]蒋忠中,汪定伟.有时间窗车辆路径问题的捕食搜索算法.控制与决策.2007,22(1):59-62页
[96] Hayashi D, Satoh T, Okita T. Distributed optimization by using artificial life.Transactions of IEE Japan.1996,116(C5):584-590P
[97] Yang B S, Lee Y H. Artificial life algorithm for function optimization. Proceedings of2000ASME Design Engineering Technical Conferences&Computers andInformation in Engineering Conference. New York, USA: ASME,2000
[98] Yang B S, Lee Y H, Choi B K, et al. Optimum design of short journal bearings byartificial life algorithm. Tribology International.2001,34:427-435P
[99] Ahn Y K, Song J D, Yang B S. Optimal design of engine mount using an artificial lifealgorithm. Journal of Sound and Vibration.2003,261:309-328P
[100]王亚钊,周永华,刘毅,等.人工生命算法的研究进展.计算技术与自动化.2006,25(4):160-163页
[101] Passino K M. Biomimicry of bacterial foraging for distributed optimization andcontrol. Control Systems, IEEE.2002,22(3):52-67P
[102] Li M S, Ji T Y, Tang W J, et al. Bacterial foraging algorithm with varying population.BioSystems.2010,100:185-197P
[103] Kumar K S, Jayabarathi T. Power system reconfiguration and loss minimization for andistribution systems using bacterial foraging optimization algorithm. Electrical Powerand Energy Systems.2012,36:13-17P
[104] Biswas A, Das S, Abraham A, et al. Stability analysis of the reproduction operator inbacterial foraging optimization. Theoretical Computer Science.2010,411:2127-2139P
[105]李晓磊,邵之江,钱积新.一种基于动物自治体的寻优模式：鱼群算法.系统工程理论与实践.2002,22(11):32-38页
[106] Shen W, Guo X P, Wu C, et al. Forecasting stock indices using radial basis functionneural networks optimized by artificial fish swarm algorithm. Knowledge-BasedSystems.2011,24:378-385P
[107] Rocha A M A C, Martins T F M C, Fernandes E M G P. An augmented Lagrangianfish swarm based method for global optimization. Journal of Computational andApplied Mathematics.2011,235:4611-4620P
[108]王联国,洪毅,赵付青,等.一种简化的人工鱼群算法.小型微型计算机系统.2009,30(8):1663-1667页
[109] Karaboga D. An idea based on honey bee swarm for numerical optimization. Kayseri:Erciyes University,2005
[110] Karaboga D, Basturk B. A powerful and efficient algorithm for numerical functionoptimization: artificial bee colony (ABC) algorithm. Journal of Global Optimization.2007,39:459-471P
[111] Karaboga D, Basturk B. On the performance of artificial bee colony (ABC) algorithm.Applied Soft Computing,2008,8:687-697P
[112] Gao W F, Liu S Y. Improved artificial bee colony algorithm for global optimization.Information Processing Letters.2011,111:871-882P
[113] Li G Q, Niu P F, Xiao X J. Development and investigation of efficient artificial beecolony algorithm for numerical function optimization. Applied Soft Computing.2012,12:320-332P
[114] Simon D. Biogeography-based optimization. IEEE Transactions on EvolutionaryComputation.2008,12(6):702-713P
[115] Ma H. An analysis of the equilibrium of migration models for biogeography-basedoptimization. Information Sciences.2010,180:3444-3464P
[116] Simon D. A dynamic system model of biogeography-based optimization. Applied SoftComputing.2011,11:5652-5661P
[117] Li X T, Wang J Y, Zhou J P, et al. A perturb biogeography based optimization withmutation for global numerical optimization. Applied Mathematics and Computation.2011,218:598-609P
[118]马海平,李雪,林升东.生物地理学优化算法的迁移率模型分析.东南大学学报(自然科学版).2009,39(I):16-21页
[119] Avriel M. Nonlinear programming: analysis and methods. Dover Publishing,2003
[120] Nocedal J, Wright S J. Numerical optimization. Springer,2006
[121] Snyman, J A. Practical mathematical optimization: an introduction to basicoptimization theory and classical and new Gradient-Based algorithms. SpringerPublishing,2005
[122]黄友锐.智能优化算法及其应用.北京:国防工业出版社,2008
[123] Shen J H, Li Y, An optimization algorithm based on optical principles. Advances inSystems Science and Applications.2009,9(4):647-655P
[124] Shen J H, Li Y. Light ray optimization and its parameter analysis. Proceeding of the2009International Joint Conference on Computational Sciences and Optimization.Sanya, China,2009:918-922P
[125]沈继红,李焱.基于正六边形网格的光线寻优算法.中国运筹学会成立三十周年暨2010学术年会.北京,中国,2010:21-22页
[126] Shen J H, Li Y. Light ray optimization with function transform. The8th InternationalConference on Optimization: Techniques and Applications. Shanghai, China,2010:21-22P
[127] Shen J H, Li J L. The principle analysis of light ray optimization algorithm.2010Second International Conference on Computational Intelligence and NaturalComputing. Wuhan, China,2010:154-157P
[128] Shen J H, Li J L. Light ray optimization algorithm and convergence analysis for onedimensional problems.2011International Conference on Fuzzy Systems and NeuralComputing. Hong Kong, China,2011:103-106P
[129]沈继红,李加莲.光线寻优算法在函数优化中的应用.控制工程.2011,18(6):845-847,850页
[130] Shen J H, Li J L. Light ray optimization algorithm based on annealing strategy.Advanced Materials Research.2012,461:435-439P
[131] Wolpert D H, Macready W G. No free lunch theorems for optimization. IEEETransactions on Evolution Compution.1997,1(1):67-82P
[132] Wolf K B. Geometry and dynamics in refracting systems. European Journal of Physics.1995,16:14-20P
[133] Rashed R. A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses. Isis.1990,81(3):464-491P
[134] Kwan A, Dudley J, Lantz E. Who really discovered snell’s Law. Physics World.2002,15(2):64P
[135] Ross J. Fermat and Least-Time: Descartes did not see the light. Executive IntelligenceReview.2008,35(15):53-55P
[136]牛文元.费马最小作用原理的社会优化价值.上海理工大学学报.2011,33(4):307-311页
[137] Schuster A. An introduction to the theory of optics. London: Edward Arnold,1904
[138]马科斯﹒波恩,埃米尔﹒沃耳夫.杨葭荪译.光学原理.第七版上册.北京:电子工业出版社,2006:98-119页
[139]宫彦军,王喜昌,宋东草,等.折射定律与反射定律的推导.物理实验.2005,25(2):41-43页
[140]洪正平.变分法在光学中的应用.山东师范大学学报(自然科学版).2003,18(2):93-94页
[141] LaRue R H, Sloman K N, Weiss M J, et al. Correspondence between traditionalmodels of functional analysis and a functional analysis of manding behavior. Researchin Developmental Disabilities.2011,32:2449-2457P
[142]徐建平,桂子鹏.变分方法.上海:同济大学出版社,1999:103-151页
[143] Agrawal O P. Formulation of Euler-Lagrange equations for fractional variationalproblems. Journal of Mathematical Analysis and Applications.2002,272(1):368-379P
[144] Agrawal O P. Generalized variational problems and Euler-Lagrange equations.Computers and Mathematics with Applications.2010,59(5):1852-1864P
[145]刘得森.变折射率介质理论及其技术实践.重庆:西南师范大学出版社,2005:1-31页.
[146] Bóna A, Slawinski M A. Fermat’s principle for seismic rays in elastic media. Journalof Applied Geophysics.2003,54(3-4):445-451P
[147] Wu M. A convergence theorem for the Newton-like methods under some kind of weakLipschitz conditions. Journal of Mathematical Analysis and Applications.2008,339:1425-1431P
[148]高全禄.二阶偏微分方程解的唯一性和稳定性.西北建筑工程学院学报.1985(1):67-75页
[149]王锐利,林大志.一阶常微分方程解的唯一性问题研究.华北水利水电学院学报.2010,31(6):153-155页
[150]李荣华,刘播.微分方程数值解法.第4版.北京:高等教育出版社,2009
[151]吴强,熊志刚.基于龙格—库塔方法的一阶微分方程组的初值问题.数学理论与应用.2006,26(3):94-97页
[152]杨淑芳,吴强. F初始状态下一阶微分方程组的数值解法.数学理论与应用.2000,20(2):64-67页
[153] Anjidani M, Effati S. Steepest descent method for solving zero-one nonlinearprogramming problems. Applied Mathematics and Computation.2007,193:197-202P
[154] Saeidi S. Modified hybrid steepest-descent methods for variational inequalities andfixed points. Mathematical and Computer Modelling.2010,52:134-142P
[155] Bousquet P, Clarke F. Local Lipschitz continuity of solutions to a problem in thecalculus of variations. Journal of Differential Equations.2007,243(2):489-503P
[156] Clarke F. Continuity of solutions to a basic problem in the calculus of variations.Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V.2005,4(3):511-530P
[157] Giaquinta M, Hildebrandt S. Calculus of Variations II. Springer,1997
[158] Razzaghi M, Yousefi S. Legendre wavelets method for the solution of nonlinearproblems in the calculus of variations. Mathematical and Computer Modelling.2001,34(1):45-54P
[159] Sagan H. Introduction to the calculus of variations. McGraw-Hill Book Company,1993
[160] Tatari M, Dehghan M. Solution of problems in calculus of variations via He’svariational iteration method. Physics Letters A.2007,362:401-406P
[161] Rosenblueth J F, Licea G S. A new sufficiency theorem for strong minima in thecalculus of variations. Applied Mathematics Letters.2005,18(11):1239-1246P
[162]钟集.高等几何.北京:高等教育出版社,1983
[163]老大中.变分法基础.第二版.北京:国防工业出版社,2007:45-131页
[164]梅加强.数学分析.北京:高等教育出版社,2011
[165] Geng X T, Xu J, Xiao J H, et al. A simple simulated annealing algorithm for themaximum clique problem. Information Sciences.2007,177:5064-5071P
[166]徐培德,邱涤珊.非线性最优化方法及应用.长沙:国防科技大学出版社,2007页
[167]袁亚湘.非线性优化计算方法.北京:科学出版社,2007:119-123页
[168] Yao X, Liu Y, Lin G. Evolutionary programming made faster. IEEE Transactions onEvolutionary Computation.1999,3(2):82-102P
[169]石鸿雁,苏晓明.实用智能优化方法.大连:大连理工大学出版社,2009:265-270页
[170] T rn A, Zilinskas A. Global optimization (Lecture Notes in Computer Science, vol.35). Berlin, Germany: Springer-Verlag,1989
[171] Schwefel H P. Evolution and optimum seeking. New York: Wiley,1995
[172]康立山,谢云,尤矢勇.非数值并行算法.第一册.模拟退火算法.北京:科学出版社,1994
[173] Majumdar J, Bhunia A K. Elitist genetic algorithm for assignment problem withimprecise goal. European Journal of Operational Research.2007,177:684-692P
[174]纪震,廖惠连,吴青华.粒子群算法及应用.北京:科学出版社,2009
[2]陈宝林.最优化理论与算法.北京:清华大学出版社,2005:1-4页
[3] Abbasbandy S, Jafarian A. Steepest descent method for solving fuzzy nonlinearequations. Applied Mathematics and Computation.2006,174(1):669-675P
[4] Dai Z F. Two modified HS type conjugate gradient methods for unconstrainedoptimization problems. Nonlinear Analysis: Theory, Method&Applications.2011,74:927-936P
[5]刘小群.优化问题的智能算法及其哲学内涵.科学技术与辩证法.2007,24(6):92-102页
[6]王凌.智能优化算法及其应用.北京:清华大学出版社&施普林格出版社,2000:1-12页
[7]刑文训,谢金星.现代优化计算方法.北京:清华大学出版社,2005:13-27页
[8] Paszkowicz W. Properties of a genetic algorithm equipped with a dynamic penaltyfunction. Computational Materials Science.2009,45:77-83P
[9] Liang Y, Leung K S. Genetic algorithm with adaptive elitist-population strategies formultimodal function optimization. Applied Soft Computing.2011,11(2):2017-2034P
[10]刘道华,原思聪,邬长安,等.面向对象的改进遗传算法优化设计.华中科技大学学报(自然科学版).2008,36(7):89-92页
[11] Cai W W, Ewing D J, Ma L. Investigation of temperature parallel simulated annealingfor optimizing continuous functions with application to hyperspectral tomography.Applied Mathematics and Computation.2011,217(12):5754-5767P
[12] Saraiva J T, Pereira M L, Mendes V T, et al. A simulated annealing based approach tosolve the generator maintenance scheduling problem. Electric Power SystemsResearch.2011,81:1283-1291P
[13] Leung S C H, Zhang D F, Zhou C L, et al. A hybrid simulated annealing metaheuristicalgorithm for the two-dimensional knapsack packing problem. Computers&Operations Resarch.2012,39:64-73P
[14]印峰,王耀南,刘炜,等.个体速度差异的蚁群算法设计及仿真.智能系统学报.2009,4(6):528-533页
[15] Puris A, Bello R, Herrera F. Analysis of the efficacy of a Two-Stage methodology forant colony optimization: Case of study with TSP and QAP. Expert Systems withApplications.2010,37:5443-5453P
[16] Xiao J, Li L P. A hybrid ant colony optimization for continuous domains. ExpertSystems with Applications.2011:38(9):11072-11077P
[17] Chen C C. Two-layer particle swarm optimization for unconstrained optimizationproblems. Applied Soft Computing.2011,11:295-304P
[18] Thangaraj R, Pant M, Abraham A, et al. Particle swarm optimization: hybridizationperspectives and experimental illustrations. Applied Mathematics and Computation.2011,217:5208-5226P
[19]傅阳光,周成平,丁明跃.标准粒子群优化算的收敛性分析.应用数学.2011,24(1):187-194P
[20]沈继红.基于光学原理的优化方法的研究.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
[21] Dantzig G B. Linear programming under uncertainty. Management Science.1955,1(3):197-206P
[22] Dantzig G B, Orden G B A, Wolfe P. Note on linear programs. Pacific J Math.1955,1(5):183-195P
[23]陈军斌,杨悦.最优化方法.北京:中国石化出版社,2011:15-37页
[24]汪定伟,王俊伟,王洪峰,等.智能优化方法.北京:高等教育出版社,2007:2-7页
[25] Jiang M, Luo Y P, Yang S Y. Stochastic convergence analysis and parameter selectionof the standard particle swarm optimization algorithm. Information Processing Letters.2007,102:8-16P
[26] Liang Y C, Zhou C G, Wang Z S. An equivalent genetic algorithm based on extendedstrings and its convergence analysis. Information Sciences.2001,138:119-135P
[27] Chakraborty P, Das S, Roy G G, et al. On convergence of the multi-objective particleswarm optimizers. Information Sciences.2011,181:1411-1425P
[28] Schmitt L M. Theory of Genetic Algorithms II: models for genetic operators over thestring-tensor representation of populations and convergence to global optima forarbitrary fitness function under scaling. Theoretical Computer Science.2004,310:181-231P
[29] Xu B L, Zhu J H, Xu H G. An ant stochastic decision based particle filter and itsconvergence. Signal Processing.2010,90:2731-2748P
[30] Meise C. On the convergence of parallel simulated annealing. Stochastic Processesand their Applications.1998,76:99-115P
[31]张慧斌,王鸿斌,胡志军.基于定常线性迭代法的PSO算法收敛性分析.计算机工程与应用.2011,47(31):35-37页
[32]沈继红,王侃,胡波.粒子群优化算法的数学基础.武汉大学学报(理学版).2011,57(4):360-364页
[33]刘星,张雪峰,蔡勇,等.改进遗传算法求解焊位群完全遍历.机械与电子.2011(3):24-26页
[34]李军华,黎明.噪声环境下遗传算法的收敛性和收敛速度估计.电子学报.2011,39(8):1898-1902页
[35]申元霞,王国胤,曾传华.相关性粒子群优化模型.软件学报.2011,22(4):695-708页
[36]高尚,杨静宇.最短路的蚁群算法收敛性分析.科学技术与工程.2006,6(3):273-277页
[37] Wang L, Tang D B. An improved adaptive genetic algorithm based on hormonemodulation mechanism for job-shop scheduling problem. Expert Systems withApplications.2011,38:7243-7250P
[38] Wu P F, Gao L Q, Zou D X, et al. An improved particle swarm optimization algorithmfor reliability problems. ISA Transactions.2011,50:71-81P
[39] Wang Z J. Improved ant colony algorithm and its simulation study. Energy Procedia.2011,11:4587-4592P
[40] Giovanni L D, Pezzella F. An improved genetic algorithm for the distributed andflexible job-shop scheduling problem. European Journal of Operational Research.2010,200:395-408P
[41] Han Y F, Shi P F. An improved ant colony algorithm for fuzzy clustering in imagesegmentation. Neurocomputing.2007,70:665-671P
[42] Gao W F, Liu S Y. Improved artificial bee colony algorithm for global optimization.Information Processing Letters.2011,111:871-882P
[43] Zhang K, Xu J, Geng X T, et al. Improved taboo search algorithm for designing DNAsequences. Progress in Natural Science.2008,18:623-627P
[44] Jiang Y, Hu T S, Huang C C, et al. An improved particle swarm optimizationalgorithm. Applied Mathematics and Computation.2007,193:231-239P
[45] Shintemirov A, Tang W J, Tang W H, et al. Improved modelling of power transformerwinding using bacterial swarming algorithm and frequency response analysis. ElectricPower Systems Research.2010,80:1111-1120P
[46] Yang J G, Zhuang Y B. An improved ant colony optimization algorithm for solving acomplex combinatorial optimization problem. Applied Soft Computing.2010,10:653-660P
[47] Wang Y, Zhou J Z, Qin H, et al. Improved chaotic particle swarm optimizationalgorithm for dynamic economic dispatch problem with valve-point effects. EnergyConversion and Management.2010,51:2893-2900P
[48] Shelokar P S, Siarry P, Jayaraman V K, et al. Particle swarm and ant colonyalgorithms hybridized for improved continuous optimization. Applied Mathematicsand Computation.2007,188:129-142P
[49] Katagiri H, Hayashida T, Nishizaki I, et al. A hybrid algorithm based on tabu searchand ant colony optimization for k-minimum spanning tree problems. Expert Systemswith Applications.2012,39:5681-5686P
[50] Kim D H, Abraham A, Cho J H. A hybrid genetic algorithm and bacterial foragingapproach for global optimization. Infromation Sciences.2007,177:3918-3937P
[51] Mhamdi B, Grayaa K, Aguili T. Hybrid of genetic algorithm with particle swarmoptimization to shape reconstruction of perfect conducting cylinders. InternationalJournal of Electronics and Communications.2011,65:1032-1039P
[52] Kao Y T, Zahara E. A hybrid genetic algorithm and particle swarm optimization formultimodal functions. Applied Soft Computing.2008,8:849-857P
[53] Sun J P, Fang L, Han J. Optimization of concrete hollow brick using hybrid geneticalgorithm combining with artificial neural networks. International Journal of Heat andMass Transfer.2010,53:5509-5518P
[54] Kiran M S, zceylan E, Gündüz M, et al. A novel hybrid approach based on particleswarm optimization and ant colony algorithm to forecast energy demand of Turkey.Energy Conversion and Management.2012,53:75-83P
[55] Marinakis Y, Marinaki M. A hybrid genetic-particle swarm optimization algorithm forthe vehicle routing problem. Expert Systems with Applications.2010,37:1446-1455P
[56] Rasmussen T K, Krink T. Improved Hidden Markov Model training for multiplesequence alignment by a particle swarm optimization-evolutionary algorithm hybrid.BioSystems.2003,72:5-17P
[57] Lee K S, Geem Z W. A new meta-heuristic algorithm for continuous engineeringoptimization: harmony search theory and practice. Computer methods in appliedmechanics and engineering.2005,194:3902-3933P
[58] Fotakis D, Sidiropoulos E. A new multi-objective self-organizing optimizationalgorithm (MOSOA) for spatial optimization problems. Applied Mathematics andComputation.2012,218:5168-5180P
[59] Farasat A, Menhaj M B, Mansouri T, et al. ARO: A new model-free optimizationalgorithm inspired from asexual reproduction. Applied Soft Computing.2010,10:1284-1292P
[60] Yildiz A R. A new design optimization framework based on immune algorithm andTaguchi’s method. Computers in Industry.2009,60:613-620P
[61] Pan W T. A new fruit fly optimization algorithm: taking the financial distress model asan example. Knowledge-Based Systems.2012,26:69-74P
[62] Lebaal N, Nouari M, Ginting A. A new optimization approach based on Kriginginterpolation and sequential quadratic programming algorithm for end millingrefractory titanium alloys. Applied Soft Computing.2011,11:5110-5119P
[63] Ochoa S, Wozny G, Repke J U. A new algorithm for global optimization:Molecular-Inspired Parallel Tempering. Computers and Chemical Engineering.2010,34:2072-2084P
[64] Oftadeh R, Mahjoob M J, Shariatpanahi M. A novel meta-heuristic optimizationalgorithm inspired by group hunting of animals: Hunting search. Computers andMathematics with Applications.2010,60:2087-2098P
[65] Holland J H. Adaptation in natural and artificial systems. Michigan: University ofMichigan Press,1975
[66] Whitley D. A genetic algorithm tutorial. Statistics and Computing.1994,4:65-85P
[67] Park C H, Lee W I, Han W S, et al. Improved genetic algorithm for multidisciplinaryoptimization of composite laminates. Computers and Structures.2008,86:1894-1903P
[68] Aryanezhad M B, Hemati M. A new genetic algorithm for solving nonconvexnonlinear programming problems. Applied Mathematics and Computation.2008,199(1):186-194P
[69] Li F C, Xu L D, Jin C X, et al. Structure of Multi-Stage Composite Genetic Algorithm(MSC-GA) and its performance. Expert Systems with Applications.2011,38(7):8929-8937P
[70] Kirkpatrick S, Gelatt C D, Vecchi M P. Optimization by simulated annealing. Science.1983,220(4598):671-680P
[71] Fabian V. Simulated annealing simulated. Computers and Mathematics withApplications.1997,33:81-94P
[72] Li N, Cha J Z, Lu Y Q. A parallel simulated annealing algorithm based on functionalfeature tree modeling for3D engineering layout design. Applied Soft Computing.2010,10:592-601P
[73] Hong W C. Traffic flow forecasting by seasonal SVR with chaotic simulatedannealing algorithm. Neurocomputing.2011,74:2096-2107P
[74]朱颢东,钟勇.一种改进的模拟退火算法.计算机技术与发展.2009,19(6):32-35页
[75] Glover F. Future paths for integer programming and links to artificial intelligence.Computers and Operations Research.1986,13(5):533-549P
[76]张晓菲,张火明.基于连续函数优化的禁忌搜索算法.中国计量学院学报.2010,21(3):251-256页
[77] Fescioglu-Unver N, Kokar M M. Self controlling tabu search algorithm for thequadratic assignment problem. Computers and Industrial Engineering.2011,60(2):310-319P
[78] Dorigo M. Optimization, learning and natural algorithms. PhD thesis, Politecnico diMilano, Italie,1992
[79] Colorni A, Dorigo M, Maniezzo V. Distributed optimization by ant colonies.Proceedings of the First European Conference on Artificial Life. Paris, France:Elsevier,1992:134-142P
[80] Colorni A, Dorigo M, Maniezzo V. An investigation of some properties of an antalgorithm. Proceedings of the Parallel Problem Solving from Nature Conference.Brussels, Belgium: Elsevier Publishing,1992:509-520P
[81]姜秋霞,王中杰.混合蚁群算法的研究及其应用.装备制造技术.2008(2):36-38页
[82] Neumann F, Witt C. Ant colony optimization and the minimum spanning tree problem.Theoretical Computer Science.2010,411(25):2406-2413P
[83] Ebergart R C, Kennenedy J. Particle swarm optimization. Proceedings of IEEEInternational Conference on Neural Networks. Perth, Australia,1995:1942-1948P
[84] Eberhart R, Kennedy J. A new optimizer using particle swarm theory. SixthInternational Symposium on Micro Machine and Human Science,1995:39-45P
[85]姜海明,谢康,王亚非.按概率突跳的改进微粒群优化算法.吉林大学学报(工学版).2007,37(1):141-145页
[86]田明,刘国志.惯性权重线性调整的局部收缩微粒群算法.吉林大学学报(理学版).2008,46(1):96-100页
[87]李剑,王乘.一种改进的自适应微粒群优化算法.华中科技大学学报(自然科学版).2008,36(3):118-121P
[88] Chen M R, Li X, Zhang X, et al. A novel particle swarm optimizer hybridized withextremal optimization. Applied Soft Computing.2010,10:367-373P
[89]秦洪德,石丽丽.一种新型的被动启发式粒子群优化算法.哈尔滨工程大学学报.2010,31(10):1298-1302页
[90]刘宏达,马忠丽.均匀粒子群算法.智能系统学报.2010,5(4):336-341P
[91] Linhares A. State-space search strategies gleaned from animal behavior: a travelingsalesman experiment. Biological Cybernetic.1998,78(3):167-173P
[92] Linhares A. Preying on optima: a predatory search strategy for combinatorial problem.Proc of IEEE International Conference on Systems, Man and Cybemetics. Piscataway,NJ: IEEE Press,1998:2974-2978P
[93] Linhares A. Synthesizing a predatory search strategy for VLSI layouts. IEEE Trans onEvolutionary Computation.1999,3(2):147-152P
[94]姚世伟,陈贤,沈海鸿.带变异特征的捕食搜索算法及其在TSP问题实验研究.科学技术与工程.2011,11(27):6652-6655P
[95]蒋忠中,汪定伟.有时间窗车辆路径问题的捕食搜索算法.控制与决策.2007,22(1):59-62页
[96] Hayashi D, Satoh T, Okita T. Distributed optimization by using artificial life.Transactions of IEE Japan.1996,116(C5):584-590P
[97] Yang B S, Lee Y H. Artificial life algorithm for function optimization. Proceedings of2000ASME Design Engineering Technical Conferences&Computers andInformation in Engineering Conference. New York, USA: ASME,2000
[98] Yang B S, Lee Y H, Choi B K, et al. Optimum design of short journal bearings byartificial life algorithm. Tribology International.2001,34:427-435P
[99] Ahn Y K, Song J D, Yang B S. Optimal design of engine mount using an artificial lifealgorithm. Journal of Sound and Vibration.2003,261:309-328P
[100]王亚钊,周永华,刘毅,等.人工生命算法的研究进展.计算技术与自动化.2006,25(4):160-163页
[101] Passino K M. Biomimicry of bacterial foraging for distributed optimization andcontrol. Control Systems, IEEE.2002,22(3):52-67P
[102] Li M S, Ji T Y, Tang W J, et al. Bacterial foraging algorithm with varying population.BioSystems.2010,100:185-197P
[103] Kumar K S, Jayabarathi T. Power system reconfiguration and loss minimization for andistribution systems using bacterial foraging optimization algorithm. Electrical Powerand Energy Systems.2012,36:13-17P
[104] Biswas A, Das S, Abraham A, et al. Stability analysis of the reproduction operator inbacterial foraging optimization. Theoretical Computer Science.2010,411:2127-2139P
[105]李晓磊,邵之江,钱积新.一种基于动物自治体的寻优模式：鱼群算法.系统工程理论与实践.2002,22(11):32-38页
[106] Shen W, Guo X P, Wu C, et al. Forecasting stock indices using radial basis functionneural networks optimized by artificial fish swarm algorithm. Knowledge-BasedSystems.2011,24:378-385P
[107] Rocha A M A C, Martins T F M C, Fernandes E M G P. An augmented Lagrangianfish swarm based method for global optimization. Journal of Computational andApplied Mathematics.2011,235:4611-4620P
[108]王联国,洪毅,赵付青,等.一种简化的人工鱼群算法.小型微型计算机系统.2009,30(8):1663-1667页
[109] Karaboga D. An idea based on honey bee swarm for numerical optimization. Kayseri:Erciyes University,2005
[110] Karaboga D, Basturk B. A powerful and efficient algorithm for numerical functionoptimization: artificial bee colony (ABC) algorithm. Journal of Global Optimization.2007,39:459-471P
[111] Karaboga D, Basturk B. On the performance of artificial bee colony (ABC) algorithm.Applied Soft Computing,2008,8:687-697P
[112] Gao W F, Liu S Y. Improved artificial bee colony algorithm for global optimization.Information Processing Letters.2011,111:871-882P
[113] Li G Q, Niu P F, Xiao X J. Development and investigation of efficient artificial beecolony algorithm for numerical function optimization. Applied Soft Computing.2012,12:320-332P
[114] Simon D. Biogeography-based optimization. IEEE Transactions on EvolutionaryComputation.2008,12(6):702-713P
[115] Ma H. An analysis of the equilibrium of migration models for biogeography-basedoptimization. Information Sciences.2010,180:3444-3464P
[116] Simon D. A dynamic system model of biogeography-based optimization. Applied SoftComputing.2011,11:5652-5661P
[117] Li X T, Wang J Y, Zhou J P, et al. A perturb biogeography based optimization withmutation for global numerical optimization. Applied Mathematics and Computation.2011,218:598-609P
[118]马海平,李雪,林升东.生物地理学优化算法的迁移率模型分析.东南大学学报(自然科学版).2009,39(I):16-21页
[119] Avriel M. Nonlinear programming: analysis and methods. Dover Publishing,2003
[120] Nocedal J, Wright S J. Numerical optimization. Springer,2006
[121] Snyman, J A. Practical mathematical optimization: an introduction to basicoptimization theory and classical and new Gradient-Based algorithms. SpringerPublishing,2005
[122]黄友锐.智能优化算法及其应用.北京:国防工业出版社,2008
[123] Shen J H, Li Y, An optimization algorithm based on optical principles. Advances inSystems Science and Applications.2009,9(4):647-655P
[124] Shen J H, Li Y. Light ray optimization and its parameter analysis. Proceeding of the2009International Joint Conference on Computational Sciences and Optimization.Sanya, China,2009:918-922P
[125]沈继红,李焱.基于正六边形网格的光线寻优算法.中国运筹学会成立三十周年暨2010学术年会.北京,中国,2010:21-22页
[126] Shen J H, Li Y. Light ray optimization with function transform. The8th InternationalConference on Optimization: Techniques and Applications. Shanghai, China,2010:21-22P
[127] Shen J H, Li J L. The principle analysis of light ray optimization algorithm.2010Second International Conference on Computational Intelligence and NaturalComputing. Wuhan, China,2010:154-157P
[128] Shen J H, Li J L. Light ray optimization algorithm and convergence analysis for onedimensional problems.2011International Conference on Fuzzy Systems and NeuralComputing. Hong Kong, China,2011:103-106P
[129]沈继红,李加莲.光线寻优算法在函数优化中的应用.控制工程.2011,18(6):845-847,850页
[130] Shen J H, Li J L. Light ray optimization algorithm based on annealing strategy.Advanced Materials Research.2012,461:435-439P
[131] Wolpert D H, Macready W G. No free lunch theorems for optimization. IEEETransactions on Evolution Compution.1997,1(1):67-82P
[132] Wolf K B. Geometry and dynamics in refracting systems. European Journal of Physics.1995,16:14-20P
[133] Rashed R. A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses. Isis.1990,81(3):464-491P
[134] Kwan A, Dudley J, Lantz E. Who really discovered snell’s Law. Physics World.2002,15(2):64P
[135] Ross J. Fermat and Least-Time: Descartes did not see the light. Executive IntelligenceReview.2008,35(15):53-55P
[136]牛文元.费马最小作用原理的社会优化价值.上海理工大学学报.2011,33(4):307-311页
[137] Schuster A. An introduction to the theory of optics. London: Edward Arnold,1904
[138]马科斯﹒波恩,埃米尔﹒沃耳夫.杨葭荪译.光学原理.第七版上册.北京:电子工业出版社,2006:98-119页
[139]宫彦军,王喜昌,宋东草,等.折射定律与反射定律的推导.物理实验.2005,25(2):41-43页
[140]洪正平.变分法在光学中的应用.山东师范大学学报(自然科学版).2003,18(2):93-94页
[141] LaRue R H, Sloman K N, Weiss M J, et al. Correspondence between traditionalmodels of functional analysis and a functional analysis of manding behavior. Researchin Developmental Disabilities.2011,32:2449-2457P
[142]徐建平,桂子鹏.变分方法.上海:同济大学出版社,1999:103-151页
[143] Agrawal O P. Formulation of Euler-Lagrange equations for fractional variationalproblems. Journal of Mathematical Analysis and Applications.2002,272(1):368-379P
[144] Agrawal O P. Generalized variational problems and Euler-Lagrange equations.Computers and Mathematics with Applications.2010,59(5):1852-1864P
[145]刘得森.变折射率介质理论及其技术实践.重庆:西南师范大学出版社,2005:1-31页.
[146] Bóna A, Slawinski M A. Fermat’s principle for seismic rays in elastic media. Journalof Applied Geophysics.2003,54(3-4):445-451P
[147] Wu M. A convergence theorem for the Newton-like methods under some kind of weakLipschitz conditions. Journal of Mathematical Analysis and Applications.2008,339:1425-1431P
[148]高全禄.二阶偏微分方程解的唯一性和稳定性.西北建筑工程学院学报.1985(1):67-75页
[149]王锐利,林大志.一阶常微分方程解的唯一性问题研究.华北水利水电学院学报.2010,31(6):153-155页
[150]李荣华,刘播.微分方程数值解法.第4版.北京:高等教育出版社,2009
[151]吴强,熊志刚.基于龙格—库塔方法的一阶微分方程组的初值问题.数学理论与应用.2006,26(3):94-97页
[152]杨淑芳,吴强. F初始状态下一阶微分方程组的数值解法.数学理论与应用.2000,20(2):64-67页
[153] Anjidani M, Effati S. Steepest descent method for solving zero-one nonlinearprogramming problems. Applied Mathematics and Computation.2007,193:197-202P
[154] Saeidi S. Modified hybrid steepest-descent methods for variational inequalities andfixed points. Mathematical and Computer Modelling.2010,52:134-142P
[155] Bousquet P, Clarke F. Local Lipschitz continuity of solutions to a problem in thecalculus of variations. Journal of Differential Equations.2007,243(2):489-503P
[156] Clarke F. Continuity of solutions to a basic problem in the calculus of variations.Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V.2005,4(3):511-530P
[157] Giaquinta M, Hildebrandt S. Calculus of Variations II. Springer,1997
[158] Razzaghi M, Yousefi S. Legendre wavelets method for the solution of nonlinearproblems in the calculus of variations. Mathematical and Computer Modelling.2001,34(1):45-54P
[159] Sagan H. Introduction to the calculus of variations. McGraw-Hill Book Company,1993
[160] Tatari M, Dehghan M. Solution of problems in calculus of variations via He’svariational iteration method. Physics Letters A.2007,362:401-406P
[161] Rosenblueth J F, Licea G S. A new sufficiency theorem for strong minima in thecalculus of variations. Applied Mathematics Letters.2005,18(11):1239-1246P
[162]钟集.高等几何.北京:高等教育出版社,1983
[163]老大中.变分法基础.第二版.北京:国防工业出版社,2007:45-131页
[164]梅加强.数学分析.北京:高等教育出版社,2011
[165] Geng X T, Xu J, Xiao J H, et al. A simple simulated annealing algorithm for themaximum clique problem. Information Sciences.2007,177:5064-5071P
[166]徐培德,邱涤珊.非线性最优化方法及应用.长沙:国防科技大学出版社,2007页
[167]袁亚湘.非线性优化计算方法.北京:科学出版社,2007:119-123页
[168] Yao X, Liu Y, Lin G. Evolutionary programming made faster. IEEE Transactions onEvolutionary Computation.1999,3(2):82-102P
[169]石鸿雁,苏晓明.实用智能优化方法.大连:大连理工大学出版社,2009:265-270页
[170] T rn A, Zilinskas A. Global optimization (Lecture Notes in Computer Science, vol.35). Berlin, Germany: Springer-Verlag,1989
[171] Schwefel H P. Evolution and optimum seeking. New York: Wiley,1995
[172]康立山,谢云,尤矢勇.非数值并行算法.第一册.模拟退火算法.北京:科学出版社,1994
[173] Majumdar J, Bhunia A K. Elitist genetic algorithm for assignment problem withimprecise goal. European Journal of Operational Research.2007,177:684-692P
[174]纪震,廖惠连,吴青华.粒子群算法及应用.北京:科学出版社,2009
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