基于鞅论的灰狼优化算法全局收敛性分析

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.

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

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.