Underdamped Particle Swarm Optimization

TL;DR

Underdamped Particle Swarm Optimization (UEPS) augments classical PSO by incorporating a damped oscillatory term in the velocity update, enabling dynamic exploration and occasional overshoot beyond local optima to enhance robustness. The approach leverages underdamped vibration concepts, characterized by parameters $\omega_0$ and $\gamma$, to balance exploration and exploitation and to efficiently traverse complex search landscapes. Through mathematical formulation, pseudocode, and Python-based implementation, the paper demonstrates UEPS’s competitive performance on unconstrained benchmark functions and constrained problems such as the Rosenbrock with constraints and the pressure vessel design, often matching or surpassing established metaheuristics with modest computational effort. The work contributes a practical, physics-inspired variant of PSO with demonstrated applicability to engineering optimization tasks and provides a runnable implementation for replication and further study.

Abstract

This article presents Underdamped Particle Swarm Optimization (UEPS), a novel metaheuristic inspired by both the Particle Swarm Optimization (PSO) algorithm and the dynamic behavior of an underdamped system. The underdamped motion acts as an intermediate solution between undamped systems, which oscillate indefinitely, and overdamped systems, which stabilize without oscillation. In the context of optimization, this type of motion allows particles to explore the search space dynamically, alternating between exploration and exploitation, with the ability to overshoot the optimal solution to explore new regions and avoid getting trapped in local optima. First, we review the concept of damped vibrations, an essential physical principle that describes how a system oscillates while losing energy over time, behaving in an underdamped, overdamped, or critically damped manner. This understanding forms the foundation for applying these concepts to optimization, ensuring a balanced management of exploration and exploitation. Furthermore, the classical PSO algorithm is discussed, highlighting its fundamental features and limitations, providing the necessary context to understand how the underdamped behavior improves PSO performance. The proposed metaheuristic is evaluated using benchmark functions and classic engineering problems, demonstrating its high robustness and efficiency.

Figures (4)

• Figure 1: Types of damped vibrations.
• Figure 2: Comparison of execution time between UEPS and PSO.
• Figure 3: Problema (\ref{['rosenbrock_restric']}).
• Figure 4: Pressure vessel design problem.