Quantifying the Impact of Modules and Their Interactions in the PSO-X Framework

TL;DR

The paper investigates how modules and their interactions in the PSO-X modular framework influence optimization performance across the CEC'05 problem suite, using 1,424 PSO-X variants evaluated on 25 functions at 10 and 30 dimensions. It applies functional ANOVA to decompose performance variance into main effects and pairwise/triple interactions among modules, revealing that a small set of modules—especially randomMatrix and omega1CS, with DNPP contributing in some cases—drives most of the explainable variance. Triple interactions exist but are generally weak, and clustering of problem classes by module-effect profiles shows broad similarity across problems with a few distinct patterns. The findings offer practical guidance for PSO-X design and selection, suggesting a portfolio strategy centered on key modules to cover diverse problem classes while acknowledging limits of the current benchmark and module set.

Abstract

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.

Figures (8)

• Figure 1: Performance distribution of 1,424 PSO-X variants on 25 problem classes comprising the CEC'05 suite. The performance values (i.e., the absolute difference between the best-found solution (within $5,000D$ evaluations) and the global optimum) are presented on a logarithmic scale.
• Figure 2: Number of effects vs. cumulative variance contribution of the 92 module effects on each of the 25 problem classes belonging to the CEC'05 suite.
• Figure 3: Performance results from the tuning of the HC algorithm. Silhouette coefficient is used as clustering quality measure.
• Figure 4: Dendrogram of the clustering process of HC. It visualizes how problem classes are iteratively grouped based on similarity, with branch heights indicating inter-cluster dissimilarity. Color coding highlights the resulting clusters.
• Figure 5: Clustermap of the 10$D$ problems.