关键词: 粒子群优化（PSO）, 离散空间, 粒子轨迹, 理论研究, 收敛

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