复杂优化问题中小生境粒子群优化算法的改进及研究
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摘要
在现实社会的生产实践中,大量实际问题的解决最终可转化为对问题的优化。而这些待优化的问题往往非常复杂,集中表现在多模值、维数高、多目标和动态性等方面。传统的进化算法受限于自身的机理和结构单一,收敛精度低、对初值敏感、易陷入局部最优,对高维复杂问题的处理比较吃力。
由Eberhart和Kennedy教授在1995年共同提出了粒子群优化算法。它源于鱼群和鸟群的觅食行为,进而提出来的一种具有代表性的群体性智能进化算法。由于粒子群优化算法具有搜索速度快、全局搜索能力强、鲁棒性强等突出优点,所以短短几年时间,粒子群优化算法已经成为计算智能领域的新的研究热点,并被应用到许多领域之中。
随着大量科研人员对粒子群优化算法的深入研究,粒子群优化算法已被成功地用来解决静态单模优化问题。但是实际生产中的许多优化问题需转化为多模优化问题和动态优化问题。面对这些复杂的优化问题,不但要求优化算法能够迅速的、准确的找到全局最优极值点,而且要找出所有的局部最优极值点并能够及时的跟踪变化的全局最优极值点。这对于粒子群优化算法则是一种新的挑战。
现从以下三方面对本文所做的工作进行阐述：
(1)粒子群优化算法的发展经历了惯性权重线性递减、全信息、单维搜索、拓扑结构、多种群等策略改进的阶段。本文对粒子群优化算法发展过程和各种改进版本在第2章作了详细的描述和分析。
(2)多模优化问题在现实生活中非常常见。例如：路径优化、数据分析、蛋白质结构预测等。这类问题不仅要寻找一个全局最优极值点,有些场合需要同时找出其余的局部最优极值点。对于多模优化问题,经典的优化算法往往易陷入局部最优点而难以找到全局最优解,更难于找到所有的局部最优极值点。但是,基于物种形成原理的小生境技术的引入使多模优化问题的求解决逐步实现。本文将在第3、4章详细介绍小生境技术的精华之处,并测试验证基于局部搜索的小生境粒子群优化算法的优势。
(3)现实世界的许多问题中存在很多动态元素。某些变量的状态常常随着时间的变化而变化。例如：股票市场、路径规划、物流配送、投资分配等。所谓动态优化问题,优化一类问题时不仅为了获得问题的全局最优解,还要能及时的检测到环境的变化,精确的跟踪最优解随时间变化的轨迹。鉴于上述改进的小生境粒子群优化算法在多模函数成功案例,我们在第5章对其做了更深入的研究和改进,并在动态优化测试函数上进行实验分析,结果证实改进算法的有效性。
Nowadays, in the real society, a large number of practical problems can be treated as optimization problem. The problem which is needed to be optimized is always complex. The complexity mainly consists of multi-modal, high dimension, multi-objective or dynamic, etc. The traditional evolutionary algorithm is limited by its own mechanism and single structure which results in low convergence precision, sensitivity to initial value, easy falling into local optimum, bad capacity of handling problems of high-dimensional complex problems.
Particle swarm optimization algorithm was proposed by Eberhart and Kennedy in1995. It is a typical swarm intelligence optimization algorithm which originated in the fish and bird flock foraging behavior. Because of the outstanding advantages of particle swarm optimization algorithm such as fast search speed, good global search ability and strong robustness, particle swarm optimization algorithm has become the new research spot in the field of computational intelligence and has been applied to many fields.
Along with more deep researches about particle swarm optimization algorithm, the algorithm has been successfully applied to solve the static single modal optimization problem. Many optimization problems in the actual production need to be transferred to multi-modal optimization and dynamic optimization problem. These complex optimization problems require optimization algorithm to find the global optimal extreme value point quickly and accurately, to find out all the local optimal extreme value points and track the changes of the global optimal extreme value point in time. This is a new challenge for particle swarm optimization algorithm.
This thesis expounds the work from the following three aspects:
(1) The development of particle swarm optimization algorithm consists of the linear decreasing inertia weight, full information, one-dimensional search, topology structure, and multi-swarm strategy, etc. In this thesis, the development process and improved versions of particle swarm optimization algorithm are described and analyzed in chapter2.
(2) Multi-modal optimization problems are very common in real life such as the path planning, data analysis and prediction of protein structure, etc. This kind of problem not only needs a global optimal extreme value point, but also needs the rest of the local optimum extreme points in some cases. For multi-modal optimization problem, a classical optimization algorithm often falls into local optimal point, so it is difficult to find the global optimal solution, not to mention all local optimal extreme value points. Then, the niche technology which is based on the principle of speciation is introduced to solve the multi-modal optimization problem gradually. The essence of the niche technology is introduced in chapter3and4in detail. And the advantages of niching particle swarm optimization algorithm based on local search was tested and verified.
(3) In fact, many problems consist of a lot of dynamic elements. Some variables often vary over time such as the stock market, path planning, logistics allocation, investment allocation, etc. Dynamic optimization problem is the class of problems which are not only intended to obtain the global optimal solution of a problem, but also has the ability to detect changes in the environment timely and track the trajectory of optimal solution which changes over time precisely. Considering that the improved niche particle swarm optimization algorithm discussed above has succeeded in multi-modal function, we did some further research and improvement in chapter5. Some experiments are conducted on dynamic optimization test functions in order to analyze the performance of the algorithm. The results prove the efficiency of the improved algorithm.
由Eberhart和Kennedy教授在1995年共同提出了粒子群优化算法。它源于鱼群和鸟群的觅食行为,进而提出来的一种具有代表性的群体性智能进化算法。由于粒子群优化算法具有搜索速度快、全局搜索能力强、鲁棒性强等突出优点,所以短短几年时间,粒子群优化算法已经成为计算智能领域的新的研究热点,并被应用到许多领域之中。
随着大量科研人员对粒子群优化算法的深入研究,粒子群优化算法已被成功地用来解决静态单模优化问题。但是实际生产中的许多优化问题需转化为多模优化问题和动态优化问题。面对这些复杂的优化问题,不但要求优化算法能够迅速的、准确的找到全局最优极值点,而且要找出所有的局部最优极值点并能够及时的跟踪变化的全局最优极值点。这对于粒子群优化算法则是一种新的挑战。
现从以下三方面对本文所做的工作进行阐述：
(1)粒子群优化算法的发展经历了惯性权重线性递减、全信息、单维搜索、拓扑结构、多种群等策略改进的阶段。本文对粒子群优化算法发展过程和各种改进版本在第2章作了详细的描述和分析。
(2)多模优化问题在现实生活中非常常见。例如：路径优化、数据分析、蛋白质结构预测等。这类问题不仅要寻找一个全局最优极值点,有些场合需要同时找出其余的局部最优极值点。对于多模优化问题,经典的优化算法往往易陷入局部最优点而难以找到全局最优解,更难于找到所有的局部最优极值点。但是,基于物种形成原理的小生境技术的引入使多模优化问题的求解决逐步实现。本文将在第3、4章详细介绍小生境技术的精华之处,并测试验证基于局部搜索的小生境粒子群优化算法的优势。
(3)现实世界的许多问题中存在很多动态元素。某些变量的状态常常随着时间的变化而变化。例如：股票市场、路径规划、物流配送、投资分配等。所谓动态优化问题,优化一类问题时不仅为了获得问题的全局最优解,还要能及时的检测到环境的变化,精确的跟踪最优解随时间变化的轨迹。鉴于上述改进的小生境粒子群优化算法在多模函数成功案例,我们在第5章对其做了更深入的研究和改进,并在动态优化测试函数上进行实验分析,结果证实改进算法的有效性。
Nowadays, in the real society, a large number of practical problems can be treated as optimization problem. The problem which is needed to be optimized is always complex. The complexity mainly consists of multi-modal, high dimension, multi-objective or dynamic, etc. The traditional evolutionary algorithm is limited by its own mechanism and single structure which results in low convergence precision, sensitivity to initial value, easy falling into local optimum, bad capacity of handling problems of high-dimensional complex problems.
Particle swarm optimization algorithm was proposed by Eberhart and Kennedy in1995. It is a typical swarm intelligence optimization algorithm which originated in the fish and bird flock foraging behavior. Because of the outstanding advantages of particle swarm optimization algorithm such as fast search speed, good global search ability and strong robustness, particle swarm optimization algorithm has become the new research spot in the field of computational intelligence and has been applied to many fields.
Along with more deep researches about particle swarm optimization algorithm, the algorithm has been successfully applied to solve the static single modal optimization problem. Many optimization problems in the actual production need to be transferred to multi-modal optimization and dynamic optimization problem. These complex optimization problems require optimization algorithm to find the global optimal extreme value point quickly and accurately, to find out all the local optimal extreme value points and track the changes of the global optimal extreme value point in time. This is a new challenge for particle swarm optimization algorithm.
This thesis expounds the work from the following three aspects:
(1) The development of particle swarm optimization algorithm consists of the linear decreasing inertia weight, full information, one-dimensional search, topology structure, and multi-swarm strategy, etc. In this thesis, the development process and improved versions of particle swarm optimization algorithm are described and analyzed in chapter2.
(2) Multi-modal optimization problems are very common in real life such as the path planning, data analysis and prediction of protein structure, etc. This kind of problem not only needs a global optimal extreme value point, but also needs the rest of the local optimum extreme points in some cases. For multi-modal optimization problem, a classical optimization algorithm often falls into local optimal point, so it is difficult to find the global optimal solution, not to mention all local optimal extreme value points. Then, the niche technology which is based on the principle of speciation is introduced to solve the multi-modal optimization problem gradually. The essence of the niche technology is introduced in chapter3and4in detail. And the advantages of niching particle swarm optimization algorithm based on local search was tested and verified.
(3) In fact, many problems consist of a lot of dynamic elements. Some variables often vary over time such as the stock market, path planning, logistics allocation, investment allocation, etc. Dynamic optimization problem is the class of problems which are not only intended to obtain the global optimal solution of a problem, but also has the ability to detect changes in the environment timely and track the trajectory of optimal solution which changes over time precisely. Considering that the improved niche particle swarm optimization algorithm discussed above has succeeded in multi-modal function, we did some further research and improvement in chapter5. Some experiments are conducted on dynamic optimization test functions in order to analyze the performance of the algorithm. The results prove the efficiency of the improved algorithm.
引文
[1]Kennedy J, Eberhart R. Particle swarm optimization. Proceedings of the IEEE International Conference on Neural Networks. Perth, Australia, Piscataway, NJ.1995,4. page:1942-1948.
[2]Goldberg D E. Genetic Algorithms in Search. Optimization & Machine Learning Addison-Wesley, Reading MA.1989
[3]Dorigo M, Maniezzo V. The ant system:optimization by a colony of cooperating agents. IEEE Transaction on System, Man and Cybernetics-part B.1996. page:1-13.
[4]Stom R, Price K. Differential evolution —simple and efficient adaptive scheme for global optimization over continuous spaces. Berkeley:University of California.2006.
[5]Xin Yao.A fuzzy clustering based niching approach to multi-modal function optimization[J]. Journal of Cognitive System Research.2000,1. page:119～133
[6]Huang Zhangcan, Wang Zongyue, Cheng Hao. Similitude Frame Algorithm for Multimodal Function Optimization. Progress in Intelligence Computation and Applications.2005.
[7]P N Suganthan. Particle swarm optimiser with neighbourhood operator Evolutionary Computation. CEC 99. Proceedings of the 1999 Congress on Volume 3.1999.
[8]J J Liang, P. N. Suganthan. Dynamic multi-swarm particle swarm optimizer Swarm Intelligence Symposium.2005. SIS 2005. Proceedings 2005 IEEE, pages:124-129.2005/6/8
[9]X Li. A multimodal particle swarm optimizer based on fitness Euclidean-distance ration. Proc. of Genetic and Evolutionary Computation Conference. pages:78-85,2007.
[10]J. P. Li, M. E. Balazs, G T. Parks. A species conserving genetic algorithm for multimodal function optimization. Evolution Computation.. page:207-234.2002.
[11]Srinivas M, Patnaik L M. Adaptive probabilities of crossover and mutation in genetic algorithms. IEEE Transactions on Systems Man and Cybernetics.1994.
[12]HAO Zhifeng, GUO Guanghan, HUANG Han. A particle swarm optimization algorithm with differential evolution. Proceedings of the Sixth International Conference on Machine Learning and Cybernetics.2007.
[13]CHEN Ye, ZHAO Guo-bo, LIU Jun-yong. An Ant Colony Optimization and Particle Swarm Optimization Hybrid Algorithm for Unit Commitment Based on Operate Coding[J].Power System Technology.2008.32(6)page:52-56.
[14]Eberhart R C, J. Kennedy. A new optimizer using particle swarm theory. Proc. of the 6th Int. Symposium on Micro Machine and Human Science. pp.39-43.1995.
[15]C. Li, S. Yang, P. N. Suganthan. Benchmark Generator for CEC'2009 Competition on Dynamic Optimization. Technical Report 2008. Department of Computer Science, University of Leicester, U.K.2008.
[16]Yang Ming, Kang Li shan, WU Yanlai. Measuring Dynamic Degree and Conflict Degree Between Objectives for Dynamic Multi-Objective TSP. Geometrics and Information Science of Wuhan University.2008.02.
[17]Zeng Jianchao and Cui Zhihua. A Guaranteed Global Convergence Particle Swarm Optimizer Journal of Computer Research and Development.2004,08.
[18]Chen Jie, Pan Feng, Wang Guanghui. Review of the PSO research in dynamic environments CAAI Transactions on Intelligent Systems.2009,03.
[19]曾三友,龙浩求.基于动态多目标演化算法的天线设计.中国宇航学会深空探测技术专业委员会,第六届学术年会暨863计划“深空探测与空间实验技术”重大项目学术研讨会.中国海南三亚。2009.12.01.
[20]Changhe Li and Shengxiang Yang. A Clustering Particle Swarm Optimizer for Dynamic Optimization.2009 IEEE Congress on Evolutionary Computation (CEC 2009).2009.
[21]Mendes Rui. DynDE:A differential evolution for dynamic optimization problems. Evolutionary Computation,2005. The 2005 IEEE Congress on. page:2808-2815.Vol.3.
[22]HU X, EBERHART R C. Adaptive particle swarm optimization:detection and response to dynamic systems. Proceedings of the IEEE Congress on Evolutionary Computation (CEC2002). Honolulu, USA,2002.
[23]CARL.ISLE.A,DOZIER.G. Adapting particle swarm optimization to dynamic environments. Proceedings of International Conference on Artificial Intelligence. Las Vegas, USA,2000.
[24]BLACKWELL T M, PETER J B. Dynamic search with charged swarms. Proceedings of the Genetic and Evolutionary Computation Conference. San Francisco, USA:Morgan Kaufmann Publishers Inc.2002,06.
[25]M. Clerc. The swarm and the queen:towards a deterministic and adaptive particle swarm optimization. In Proceedings of the IEEE Congress on Evolutionary Computation (CEC 1999). pp:1951-1957.1999.
[26]M. Clerc and J. Kennedy. The particle swarm-explosion, stability and convergence in a multidimensional complex space [J]. IEEE Transactions on Evolutionary Computation, vol. 6. page:58-73.2002.
[27]Shi Y, Eberhart R C. A modified swarm optimizer[C]. IEEE International Conference of Evolutionary Computation.1998.
[28]Suganthan P. N. Particle swarm optimizer with neighborhood operator. Proceedings of the IEEE congress on evolutionary computation (CEC).1999.
[29]Mendes R, Kennedy J, Neves J. The fully informed particle swarm:simpler, maybe better. IEEE Transactions on Evolutionary Computation.2004.
[30]Liang J, Qin A, Suganthan P N. Comprehensive learning particle swarm optimizer for global optimization of multi-modal functions[J].IEEE Transactions on Evolutionary Computation.2006.
[31]J J Liang, PN Suganthan. Dynamic multi-swarm particle swarm optimizerSwarm Intelligence Symposium [J]. SIS 2005. Proceedings 2005 IEEE. Pages:124-129
[32]MIN Ke-xue, GE Hong-wei, ZHANG Yi. Solving Traveling Salesman Problems by an ACO-and-PSO-Based Hybrid Algorithm. Journal of Jilin University(Information Science Edition).2006.04.
[33]LIU Jing-jing, WU Chuan-sheng. A Modified Particle Swarm Optimization Algorithm with Crossover Operator. Journal of Qingdao University of Science and Technology(Natural Science Edition).2008.01
[34]Gao Shang, Yang Jingyu,Wu Xiaojun. Particle Swarm Optimization Based on the ideal of simulated annealing algorithm [J]. Computer Applications and Software.2005,01.
[35]Eberhart R C and Shi Y. Particle swarm optimization:development, applications and resources. Proceeding of congress on Evolutionary Computation.2001.
[36]K. Koper and M. Wysession. Multimodal function optimization with a niching genetic algorithm:A seismological example. Bulletin of the Seismological Society of America, vol. 89. pp:978-988.1999.
[37]R. Storn and K. V. Price. Differential evolution-A simple and efficient heuristic for global optimization over continuous spaces. J. of Global Optimization, vol.11. pp.341-359.1995.
[38]Liu J, Lampinen J. On setting the control parameter of the differential evolution method. Proc 8th Int Conf Soft Computing(MENDEL2002) [C].2002.
[39]Alberto Colorni, Marco Dorigo, Vittorio Maniezzo. Distributed optimization by ant colonies. Proceedings of the First European Conference on Artificial Life [C].1991.
[40]Kirkpatrick S, Vecchi MP. Optimization by simulated annealing. Science.1983.
[41]Lee C G, Cho D H, Jung H K. Niche genetic algorithm with restricted competition selection for multimodal function optimization. IEEE Transactions on Magnetics.1999
[42]Thierens D. Scalability problems of simple genetic algorithms. Evolutionary Computation[J].1999
[43]Glodberg D E, Richardson J. Genetic algorithms with sharing for multi-modall function optimization. Proc of 2nd Int Conf on Genetic Algorithms [C].1987.
[44]G. R. Harik. Finding multimodal solutions using restricted tournament selection. In proceedings of the sixth International Conference on Genetic Algorithms[C]. Morgan Kaufmann.
[45]A. Petrowski. A clearing procedure as a niching method for genetic algorithms. Proc. of the IEEE Int. Conf. on Evolutionary Computation[C]. New York, USA.1996. pp:798-803.
[46]D. Zaharie. Extensions of differential evolution algorithms for multimodal optimization. Proc of 6th Int. Symposium of Symbolic and Numeric Algorithms for Scientific Computing [J]. pp:523-534.2004.
[47]PAN Xi-jiao, ZHANG Jun. A novel adaptive sequential niche particle swarm optimization algorithm. Journal of Anhui University of Technology and Science(Natural Science)[J].2007,01.
[48]Sander, Martin Ester, Hans-Peter Kriegel. Density-Based Clustering in Spatial Databases: The Algorithm GDBSCAN and Its Applications[J].1998.
[49]Lingyun Wei, Mei Zhao. A niche hybrid genetic algorithm for global optimization of continuous multimodal functions. Journal of Applied Mathematics.2005.
[50]Wang Chunxiang, Li Xianyou. A New Niche Genetic Algorithm with(1+1) Competition Strategy Based on the Shortest Euclidean Distance[J]. Computer & Digital Engineering.2008,11.
[51]Goldberg D E, Wang L. Adaptive niching via coevolutionary sharing.1997.
[52]X. Li. A multimodal particle swarm optimizer based on fitness Euclidean-distance ration[C]. Proc. of Genetic and Evolutionary Computation Conference, pp:78-85.2007.
[53]X. Li. Adaptively choosing neighborhood bests using species in a particle swarm optimizer for multimodal function optimization[C]. In Proc of Genetic and Evolutionary Computation Conference 2004 (LNCS3102). K. Deb, Ed.. pp:105-116.2004.
[54]J. J. Liang, T. P. Runarsson, P. N. Suganthan etc. Problem Definitions and Evaluation Criteria for the CEC 2006 Special Session on Constrained Real-Parameter Optimization[C]. Technical Report, Nan yang Technological University, Singapore, March 2006.
[55]K. Veeramachaneni, T. Peram, C. Mohan. Optimization using particle swarm with near neighbor interactions[C].In Proc. of Genetic and Evolutionary Computation Conference. pp: 110-121. Chicago,2003.
[56]X. Li. Niching without niching parameters:particle swarm optimization using a ring topology [J]. IEEE Transactions on Evolutionary Computation. vol.14. pp:150-169. Feb 2010.
[57]Changhe Li and Shengxiang Yang. A Clustering Particle Swarm Optimizer for Dynamic Optimization.2009 IEEE Congress on Evolutionary Computation (CEC 2009) [C].2009.
[58]D. Parrott and X. Li. A particle swarm modall for tracking multiple peaks in a dynamic environment using speciation. Proc. of the 2004 IEEE Congress on Evolutionary Computation, pp:98-103.2004.
[59]E. L. Yu and P. N. Suganthan. Evolutionary Programming with Ensemble of Explicit Memories for Dynamic Optimization.CEC'2009 Competition on Dynamic Optimization.2009.
[2]Goldberg D E. Genetic Algorithms in Search. Optimization & Machine Learning Addison-Wesley, Reading MA.1989
[3]Dorigo M, Maniezzo V. The ant system:optimization by a colony of cooperating agents. IEEE Transaction on System, Man and Cybernetics-part B.1996. page:1-13.
[4]Stom R, Price K. Differential evolution —simple and efficient adaptive scheme for global optimization over continuous spaces. Berkeley:University of California.2006.
[5]Xin Yao.A fuzzy clustering based niching approach to multi-modal function optimization[J]. Journal of Cognitive System Research.2000,1. page:119～133
[6]Huang Zhangcan, Wang Zongyue, Cheng Hao. Similitude Frame Algorithm for Multimodal Function Optimization. Progress in Intelligence Computation and Applications.2005.
[7]P N Suganthan. Particle swarm optimiser with neighbourhood operator Evolutionary Computation. CEC 99. Proceedings of the 1999 Congress on Volume 3.1999.
[8]J J Liang, P. N. Suganthan. Dynamic multi-swarm particle swarm optimizer Swarm Intelligence Symposium.2005. SIS 2005. Proceedings 2005 IEEE, pages:124-129.2005/6/8
[9]X Li. A multimodal particle swarm optimizer based on fitness Euclidean-distance ration. Proc. of Genetic and Evolutionary Computation Conference. pages:78-85,2007.
[10]J. P. Li, M. E. Balazs, G T. Parks. A species conserving genetic algorithm for multimodal function optimization. Evolution Computation.. page:207-234.2002.
[11]Srinivas M, Patnaik L M. Adaptive probabilities of crossover and mutation in genetic algorithms. IEEE Transactions on Systems Man and Cybernetics.1994.
[12]HAO Zhifeng, GUO Guanghan, HUANG Han. A particle swarm optimization algorithm with differential evolution. Proceedings of the Sixth International Conference on Machine Learning and Cybernetics.2007.
[13]CHEN Ye, ZHAO Guo-bo, LIU Jun-yong. An Ant Colony Optimization and Particle Swarm Optimization Hybrid Algorithm for Unit Commitment Based on Operate Coding[J].Power System Technology.2008.32(6)page:52-56.
[14]Eberhart R C, J. Kennedy. A new optimizer using particle swarm theory. Proc. of the 6th Int. Symposium on Micro Machine and Human Science. pp.39-43.1995.
[15]C. Li, S. Yang, P. N. Suganthan. Benchmark Generator for CEC'2009 Competition on Dynamic Optimization. Technical Report 2008. Department of Computer Science, University of Leicester, U.K.2008.
[16]Yang Ming, Kang Li shan, WU Yanlai. Measuring Dynamic Degree and Conflict Degree Between Objectives for Dynamic Multi-Objective TSP. Geometrics and Information Science of Wuhan University.2008.02.
[17]Zeng Jianchao and Cui Zhihua. A Guaranteed Global Convergence Particle Swarm Optimizer Journal of Computer Research and Development.2004,08.
[18]Chen Jie, Pan Feng, Wang Guanghui. Review of the PSO research in dynamic environments CAAI Transactions on Intelligent Systems.2009,03.
[19]曾三友,龙浩求.基于动态多目标演化算法的天线设计.中国宇航学会深空探测技术专业委员会,第六届学术年会暨863计划“深空探测与空间实验技术”重大项目学术研讨会.中国海南三亚。2009.12.01.
[20]Changhe Li and Shengxiang Yang. A Clustering Particle Swarm Optimizer for Dynamic Optimization.2009 IEEE Congress on Evolutionary Computation (CEC 2009).2009.
[21]Mendes Rui. DynDE:A differential evolution for dynamic optimization problems. Evolutionary Computation,2005. The 2005 IEEE Congress on. page:2808-2815.Vol.3.
[22]HU X, EBERHART R C. Adaptive particle swarm optimization:detection and response to dynamic systems. Proceedings of the IEEE Congress on Evolutionary Computation (CEC2002). Honolulu, USA,2002.
[23]CARL.ISLE.A,DOZIER.G. Adapting particle swarm optimization to dynamic environments. Proceedings of International Conference on Artificial Intelligence. Las Vegas, USA,2000.
[24]BLACKWELL T M, PETER J B. Dynamic search with charged swarms. Proceedings of the Genetic and Evolutionary Computation Conference. San Francisco, USA:Morgan Kaufmann Publishers Inc.2002,06.
[25]M. Clerc. The swarm and the queen:towards a deterministic and adaptive particle swarm optimization. In Proceedings of the IEEE Congress on Evolutionary Computation (CEC 1999). pp:1951-1957.1999.
[26]M. Clerc and J. Kennedy. The particle swarm-explosion, stability and convergence in a multidimensional complex space [J]. IEEE Transactions on Evolutionary Computation, vol. 6. page:58-73.2002.
[27]Shi Y, Eberhart R C. A modified swarm optimizer[C]. IEEE International Conference of Evolutionary Computation.1998.
[28]Suganthan P. N. Particle swarm optimizer with neighborhood operator. Proceedings of the IEEE congress on evolutionary computation (CEC).1999.
[29]Mendes R, Kennedy J, Neves J. The fully informed particle swarm:simpler, maybe better. IEEE Transactions on Evolutionary Computation.2004.
[30]Liang J, Qin A, Suganthan P N. Comprehensive learning particle swarm optimizer for global optimization of multi-modal functions[J].IEEE Transactions on Evolutionary Computation.2006.
[31]J J Liang, PN Suganthan. Dynamic multi-swarm particle swarm optimizerSwarm Intelligence Symposium [J]. SIS 2005. Proceedings 2005 IEEE. Pages:124-129
[32]MIN Ke-xue, GE Hong-wei, ZHANG Yi. Solving Traveling Salesman Problems by an ACO-and-PSO-Based Hybrid Algorithm. Journal of Jilin University(Information Science Edition).2006.04.
[33]LIU Jing-jing, WU Chuan-sheng. A Modified Particle Swarm Optimization Algorithm with Crossover Operator. Journal of Qingdao University of Science and Technology(Natural Science Edition).2008.01
[34]Gao Shang, Yang Jingyu,Wu Xiaojun. Particle Swarm Optimization Based on the ideal of simulated annealing algorithm [J]. Computer Applications and Software.2005,01.
[35]Eberhart R C and Shi Y. Particle swarm optimization:development, applications and resources. Proceeding of congress on Evolutionary Computation.2001.
[36]K. Koper and M. Wysession. Multimodal function optimization with a niching genetic algorithm:A seismological example. Bulletin of the Seismological Society of America, vol. 89. pp:978-988.1999.
[37]R. Storn and K. V. Price. Differential evolution-A simple and efficient heuristic for global optimization over continuous spaces. J. of Global Optimization, vol.11. pp.341-359.1995.
[38]Liu J, Lampinen J. On setting the control parameter of the differential evolution method. Proc 8th Int Conf Soft Computing(MENDEL2002) [C].2002.
[39]Alberto Colorni, Marco Dorigo, Vittorio Maniezzo. Distributed optimization by ant colonies. Proceedings of the First European Conference on Artificial Life [C].1991.
[40]Kirkpatrick S, Vecchi MP. Optimization by simulated annealing. Science.1983.
[41]Lee C G, Cho D H, Jung H K. Niche genetic algorithm with restricted competition selection for multimodal function optimization. IEEE Transactions on Magnetics.1999
[42]Thierens D. Scalability problems of simple genetic algorithms. Evolutionary Computation[J].1999
[43]Glodberg D E, Richardson J. Genetic algorithms with sharing for multi-modall function optimization. Proc of 2nd Int Conf on Genetic Algorithms [C].1987.
[44]G. R. Harik. Finding multimodal solutions using restricted tournament selection. In proceedings of the sixth International Conference on Genetic Algorithms[C]. Morgan Kaufmann.
[45]A. Petrowski. A clearing procedure as a niching method for genetic algorithms. Proc. of the IEEE Int. Conf. on Evolutionary Computation[C]. New York, USA.1996. pp:798-803.
[46]D. Zaharie. Extensions of differential evolution algorithms for multimodal optimization. Proc of 6th Int. Symposium of Symbolic and Numeric Algorithms for Scientific Computing [J]. pp:523-534.2004.
[47]PAN Xi-jiao, ZHANG Jun. A novel adaptive sequential niche particle swarm optimization algorithm. Journal of Anhui University of Technology and Science(Natural Science)[J].2007,01.
[48]Sander, Martin Ester, Hans-Peter Kriegel. Density-Based Clustering in Spatial Databases: The Algorithm GDBSCAN and Its Applications[J].1998.
[49]Lingyun Wei, Mei Zhao. A niche hybrid genetic algorithm for global optimization of continuous multimodal functions. Journal of Applied Mathematics.2005.
[50]Wang Chunxiang, Li Xianyou. A New Niche Genetic Algorithm with(1+1) Competition Strategy Based on the Shortest Euclidean Distance[J]. Computer & Digital Engineering.2008,11.
[51]Goldberg D E, Wang L. Adaptive niching via coevolutionary sharing.1997.
[52]X. Li. A multimodal particle swarm optimizer based on fitness Euclidean-distance ration[C]. Proc. of Genetic and Evolutionary Computation Conference, pp:78-85.2007.
[53]X. Li. Adaptively choosing neighborhood bests using species in a particle swarm optimizer for multimodal function optimization[C]. In Proc of Genetic and Evolutionary Computation Conference 2004 (LNCS3102). K. Deb, Ed.. pp:105-116.2004.
[54]J. J. Liang, T. P. Runarsson, P. N. Suganthan etc. Problem Definitions and Evaluation Criteria for the CEC 2006 Special Session on Constrained Real-Parameter Optimization[C]. Technical Report, Nan yang Technological University, Singapore, March 2006.
[55]K. Veeramachaneni, T. Peram, C. Mohan. Optimization using particle swarm with near neighbor interactions[C].In Proc. of Genetic and Evolutionary Computation Conference. pp: 110-121. Chicago,2003.
[56]X. Li. Niching without niching parameters:particle swarm optimization using a ring topology [J]. IEEE Transactions on Evolutionary Computation. vol.14. pp:150-169. Feb 2010.
[57]Changhe Li and Shengxiang Yang. A Clustering Particle Swarm Optimizer for Dynamic Optimization.2009 IEEE Congress on Evolutionary Computation (CEC 2009) [C].2009.
[58]D. Parrott and X. Li. A particle swarm modall for tracking multiple peaks in a dynamic environment using speciation. Proc. of the 2004 IEEE Congress on Evolutionary Computation, pp:98-103.2004.
[59]E. L. Yu and P. N. Suganthan. Evolutionary Programming with Ensemble of Explicit Memories for Dynamic Optimization.CEC'2009 Competition on Dynamic Optimization.2009.