基于鞅论的灰狼优化算法全局收敛性分析  

Global convergence analysis of grey wolf optimization algorithm based on martingale theory

作　　者：孙丽君 冯斌斌[1,2,3] 陈天飞 SUN Li-jun;FENG Bin-bin;CHEN Tian-fei(Key Laboratory of Grain Information Processing and Control of Ministry of Education,Henan University of Technology,Zhengzhou 450001,China;Zhengzhou Key Laboratory of Machine Perception and Intelligent System,Henan University of Technology,Zhengzhou 450001,China;College of Information Science and Engineering,Henan University of Technology,Zhengzhou 450001,China)

机构地区：[1]河南工业大学粮食信息处理与控制教育部重点实验室,郑州450001 [2]河南工业大学郑州市机器感知与智能系统重点实验室,郑州450001 [3]河南工业大学信息科学与工程学院,郑州450001

出　　处：《控制与决策》2022年第11期2839-2848,共10页Control and Decision

基　　金：国家自然科学基金项目(61803146,61973104);河南省优秀青年科学基金项目(212300410036);河南省高校科技创新人才支持计划项目(21HASTIT029);河南省高等学校青年骨干教师培养计划项目(2019GGJS089);河南省青年人才托举工程项目(2019HYTP005);河南省科技攻关项目(212102210169,212102210086);河南省高等教育教学改革研究与实践项目(2019SJGLX270);河南工业大学自科创新基金支持计划项目(2020ZKCJ06);河南工业大学青年骨干教师培育计划项目(21420080)。

摘　　要：灰狼优化(grey wolf optimization,GWO)算法是一种基于群体智能的随机优化算法,已成功地应用于许多复杂的优化问题的求解.尽管GWO算法有很多改进形式,但缺少严谨的收敛性分析,导致改进后的算法不具备理论支撑.对此,运用鞅论分析其收敛性.首先,根据GWO算法原理建立其基本的数学模型,通过定义灰狼状态空间及灰狼群状态空间,建立GWO算法的Markov链模型,并分析该算法的Markov性质;其次,介绍鞅理论,推导出一个上鞅作为最优适应度值的群进化序列;然后,运用上鞅收敛定理,并结合其Markov性质对GWO算法进行收敛性分析,证明GWO算法能以1的可能性达到全局收敛;最后,通过数值实验验证其收敛性能.实验结果表明,GWO算法具有全局收敛性强、计算耗时较低、寻优精度高等特点.The grey wolf optimization(GWO)algorithm is a stochastic optimization algorithm based on swarm intelligence which has been successfully used to solve many complex optimization problems.At present,there are many improved forms of the GWO algorithm,but the lack of rigorous convergence analysis leads to no theoretical support for the improved algorithm.In order to make up for this deficiency,the martingale theory is used to analyze its convergence for the first time.Firstly,the basic mathematical model is established according to the principle of the GWO algorithm.By defining the gray wolf state space and the gray wolf group state space,the Markov chain model of the GWO algorithm is established,and the Markov properties of the algorithm are analyzed.Secondly,the martingale theory is introduced,and a swarm evolution sequence with the supermartingale as the optimal fitness value is derived.Thirdly,the convergence of the GWO algorithm is analyzed using the supermartingale convergence theorem and its Markov properties.It is proved that the GWO algorithm can achieve global convergence with the possibility of 1.Finally,the convergence performance is verified by numerical experiments.The experimental results show that the GWO algorithm has strong global convergence,low computation time and high optimization accuracy.

关 键 词：灰狼优化算法 MARKOV链 鞅论 状态转移 数值实验 全局收敛 

分 类 号：TP273[自动化与计算机技术—检测技术与自动化装置]