The PSO-X framework incorporates dozens of modules that have been proposed for solving single-objective continuous optimization problems using particle swarm optimization. While modular frameworks enable users to automatically generate and configure algorithms tailored to specific optimization problems, the complexity of this process increases with the number of modules in the framework and the degrees of freedom defined for their interaction. Understanding how modules affect the performance of algorithms for different problems is critical to making the process of finding effective implementations more efficient and identifying promising areas for further investigation. Despite their practical applications and scientific relevance, there is a lack of empirical studies investigating which modules matter most in modular optimization frameworks and how they interact. In this paper, we analyze the performance of 1424 particle swarm optimization algorithms instantiated from the PSO-X framework on the 25 functions in the CEC'05 benchmark suite with 10 and 30 dimensions. We use functional ANOVA to quantify the impact of modules and their combinations on performance in different problem classes. In practice, this allows us to identify which modules have greater influence on PSO-X performance depending on problem features such as multimodality, mathematical transformations and varying dimensionality. We then perform a cluster analysis to identify groups of problem classes that share similar module effect patterns. Our results show low variability in the importance of modules in all problem classes, suggesting that particle swarm optimization performance is driven by a few influential modules.

翻译：PSO-X框架整合了数十种模块，这些模块旨在利用粒子群优化算法解决单目标连续优化问题。尽管模块化框架使用户能够针对特定优化问题自动生成和配置算法，但这一过程的复杂性会随着框架中模块数量的增加以及模块间交互自由度的提升而加剧。理解模块如何影响算法在不同问题上的性能，对于更高效地寻找有效实现方案以及确定有前景的后续研究方向至关重要。尽管模块化优化框架具有实际应用价值和科学意义，但目前仍缺乏实证研究来探讨哪些模块最为关键以及它们如何相互作用。本文中，我们分析了基于PSO-X框架实例化的1424种粒子群优化算法在CEC'05基准测试套件中25个函数（维度分别为10和30）上的性能表现。我们采用函数方差分析（functional ANOVA）来量化模块及其组合在不同问题类别中对性能的影响。在实践中，这使我们能够根据问题的特征（如多模态性、数学变换和维度变化）识别哪些模块对PSO-X性能具有更大影响。随后，我们通过聚类分析来识别具有相似模块效应模式的问题类别组。研究结果表明，在所有问题类别中模块重要性呈现较低的变异性，这表明粒子群优化的性能主要由少数关键模块驱动。