State-space adaptive exploration for explainable particle swarm optimization

A systems theory framework for swarm optimization algorithms promises the rigorous analysis of swarm behaviors and systematic approaches that could avoid ad hoc parameter settings and achieve guaranteed performances. However, optimization processes must treat various systems theory concepts, such as stability and controllability, differently, as swarm optimization relies on preserving diversity rather than reaching uniform agent behavior. This work addresses this duality of perspective and proposes State-Space Particle Swarm Optimization (SS-PSO) using the feedback concept in control systems theory. By exploiting the hidden analogy between these two paradigms, we introduce the concept of controllability for optimization purposes through statespace representation. Extending controllability to particle swarm optimization (PSO) highlights the ability to span the search space, emphasizing the significance of particles' movement rather than their loss of diversity. Furthermore, adaptive exploration (AE) using an iterative bisection algorithm is proposed for the PSO parameters that leverages this controllability measure and its minimum singular value to facilitate explainable swarm behaviors and escape local minima. Theoretical and numerical analyses reveal that SS-PSO is only uncontrollable when the cognitive factor is zero, implying less exploration. Finally, AE enhances exploration by increasing the controllability matrix's minimum singular value. This result underscores the profound connection between the controllability matrix and exploration, a critical insight that significantly enhances our understanding of swarm optimization. AE-based State-Space-PSO (AESS-PSO) shows improved exploration and performance over PSO in 86 SOP and CEC benchmarks, particularly for smaller populations.