标题： PSO与旅行商问题：一种智能优化方法 显示英文标题

标题： PSO and the Traveling Salesman Problem: An Intelligent Optimization Approach

摘要： 旅行商问题（TSP）是一个著名的组合优化问题，旨在找到一条最短的路径，该路径依次访问每个城市一次并返回起点。 本文探讨了粒子群优化（PSO）算法在解决TSP中的应用，PSO是一种基于种群的优化算法。 尽管PSO最初是为连续优化问题设计的，但本研究通过将城市顺序视为排列来适应TSP的离散性。 在更新粒子位置后，应用了一种局部搜索策略，包括2-opt和3-opt技术，以改进解决方案。 所提出的PSO算法通过基准TSP实例进行评估，并与其他流行的优化算法如遗传算法（GA）和模拟退火（SA）进行比较。 结果表明，PSO在小到中等规模的问题上表现良好，但由于难以逃离局部最优，其在较大实例上的表现有所下降。 本文结论认为，PSO是解决TSP的一种有前景的方法，通过与其他优化技术结合可以进一步提升性能。 显示英文摘要

摘要： The Traveling Salesman Problem (TSP) is a well-known combinatorial optimization problem that aims to find the shortest possible route that visits each city exactly once and returns to the starting point. This paper explores the application of Particle Swarm Optimization (PSO), a population-based optimization algorithm, to solve TSP. Although PSO was originally designed for continuous optimization problems, this work adapts PSO for the discrete nature of TSP by treating the order of cities as a permutation. A local search strategy, including 2-opt and 3-opt techniques, is applied to improve the solution after updating the particle positions. The performance of the proposed PSO algorithm is evaluated using benchmark TSP instances and compared to other popular optimization algorithms, such as Genetic Algorithms (GA) and Simulated Annealing (SA). Results show that PSO performs well for small to medium-sized problems, though its performance diminishes for larger instances due to difficulties in escaping local optima. This paper concludes that PSO is a promising approach for solving TSP, with potential for further improvement through hybridization with other optimization techniques.