Abstract: Considering the insufficiency of the theoretical research on discrete particle swarm optimization (PSO) algorithms, this paper studies the basic theory for discrete PSO algorithms. Firstly, this paper uses order difference equations to model the generalization trajectory of particles and proposes a basic framework for the theoretical research on discrete PSO algorithms. Secondly, this paper uses the basic mathematical methods, including the limit, series convergence and Cauchy convergence theorem, to analyze the trajectory characteristics of random particle in a multi-dimension discrete space, and then provides a formal proof that each particle of a discrete PSO algorithm converges to a stable point. Finally, combined with a classical discrete optimization problem, this paper presents the empirical analysis of stochastic particles in a 15-dimensional discrete space, and the experimental results support the conclusions drawn from the theoretical findings.

Key words: particle swarm optimization (PSO), discrete space, particle trajectory, theoretical research, convergence

摘要： 当前离散粒子群优化（particle swarm optimization，PSO）算法缺乏系统性的理论支撑，因此对离散PSO算法基础理论问题进行了研究。采用一阶非齐次差分方程组对粒子群在多维离散空间的通用轨迹进行建模，为离散PSO算法的理论研究构建了基础框架；采用极限、级数收敛和柯西审敛定理等基本数学理论分析了多维离散空间随机粒子的轨迹行为特性，并证明了轨迹的收敛性；结合经典的离散优化问题，在15维离散空间中对随机粒子运动轨迹进行了实证分析，验证了理论证明的相关结果。

关键词: 粒子群优化（PSO）, 离散空间, 粒子轨迹, 理论研究, 收敛