模糊随机粒子群优化方法理论分析研究

This is an inter-disciplinary research project between information science and mathematics that focuses on the stability, convergence and the time complexity of the particle swarm optimization algorithm when fuzzy characteristics are included. The aims of this project are three-fold. First, we aim to construct a Takagi-Sugeno fuzzy particle swarm optimization algorithm and establish a general framework for its stability analysis in high-dimensional bounded space problems. Secondly, we seek to build a fuzzy random kernel for particle swarm optimization that will likely reveal the impact of the algorithm’s selection and update operations on the efficiency of the optimization process. The convergence of the particle swarm optimization in chance measure could then be further explored with the included fuzzy characteristics. The third and final objective of the project is the investigation of the time complexity properties of particle swarm optimization. Specific fuzzy random variables will be studied in chance measure and the probability distributions of the time to optimization as well as the time interval between successive iterations will be determined for some real-world problems. It is envisaged that the above results will form a strong theoretical basis for the development of fuzzy particle swarm optimization algorithms that can be applied to more diverse areas and provide new methods for other swarm intelligence optimization theoretical research.

本项目为信息科学与数学科学的交叉理论研究。粒子群优化是一种典型的群智能随机优化方法，本项目综合粒子群优化随机不确定性和在应用中不断凸显的模糊特征，基于模糊随机、Takagi-Sugeno(T-S)模糊等数学理论，研究模糊随机粒子群优化的稳定性、收敛性和时间复杂度三个问题。具体为：1）变换高维有界空间中粒子群优化迭代公式，设计高维T-S模糊粒子群优化方法，并研究其稳定性；2）构建粒子群优化的模糊随机核，研究粒子历史最优和全局最优位置的选择算子和迭代更新算子对优化过程收敛性的影响，确保模糊随机粒子群优化的测度收敛性；3）采用模糊随机变量的机会测度度量优化过程，理论推导模糊随机粒子群优化时间和迭代时间间隔的概率分布，构建模糊随机粒子群优化的时间复杂度模型。本项目的完成将基本形成一个模糊随机粒子群优化方法理论分析框架，为其在工程中更广泛的应用奠定理论基础，也为其它群智能理论研究提供新思路。

本项目基于模糊随机、Lyapunov、图论等理论，重点研究了粒子群优化稳定性、收敛性及时间复杂度等理论问题，同时开展了与神经网络系统及耦合系统的交叉研究，取得的主要研究成果有：（1）通过对粒子群优化过程中粒子随机选择解概率及粒子层级变迁概率数学建模，提出了基于模糊随机变量的群智能收敛的分析方法；（2）基于数据相似性度量方式，探讨粒子群优化在高维优化空间中度量特性，并改进粒子群优化方法将其应用于计算机配色优化；（3）研究了忆阻神经网络周期解和时变耦合系统平凡解的稳定性；（4）构造单层神经网络求解非光滑复变量实值凸优化问题，同时实现了双层二次优化问题在有限时间内求解。. 本项目的完成不仅丰富和发展了粒子群优化理论分析框架，而且为模糊粒子群优化实际应用提供理论支撑，同时提供了一个新的视角开展群智能与神经网络系统或耦合系统的交叉研究。