粒子群优化算法的数学基础

粒子群优化算法的数学基础 The Mathematical Basis of the Particle Swarm Optimization Algorithm 沈继红 SHEN Jihong1 1 first-author 王侃 WANG Kan2 2 胡波 HU Bo2 2 哈尔滨工程大学理学院 哈尔滨工程大学理学院 College of Science,Harbin Engineering University College of Science,Harbin Engineering University 哈尔滨工程大学自动化学院 哈尔滨工程大学自动化学院 College of Automation,Harbin Engineering University College of Automation,Harbin Engineering University 总结概括了粒子群优化（PSO）算法基本要素的数学描述,证明了粒子群算法解空间为赋范空间,将粒子群迭代关系定义为赋范空间的压缩映射,通过利用Banach空间的相关理论和压缩映像原理,严格证明了PSO粒子收敛位置的存在惟一性,概括了PSO算法收敛性分析的数学描述,引入概率论以及随机过程经典理论推导出PSO算法稳定的参数条件,证明了在满足此条件基础下PSO算法收敛到全局最优位置的概率为1. The general mathematical description of the Particle Swarm Optimization（PSO） is given.We prove that the solution space of PSO is normed space,and the iterative relations of PSO is the contraction mapping.The existence and the uniqueness of the PSO’s convergence position are proved by using the theorem of Banach space and the contraction mapping principles.And we summarize the mathematical description of the convergence.Using the probability theory and stochastic process we give the parameter conditions that influence the stability of PSO.At last we prove that the probability which PSO converging the position of the best of all is 1 if the parameter satisfies the condition. Banach空间 压缩映像原理 稳定性 收敛性 参数选择 Banach space contraction mapping principle stability convergence parameter selection 10.14188/j.1671-8836.2011.04.011 TP301.6 黑龙江省自然科学基金资助项目（F200931） 2011-04-01 2021-04-01 4 57