Explanation of Stagnation at Points that are not Local Optima in Particle Swarm Optimization by Potential Analysis

TL;DR

This paper analyzes why particle swarm optimization (PSO) can stagnate without reaching local optima, using potential analysis to identify conditions under which the particles lose relevance in certain dimensions, leading to non-convergence.

Contribution

The paper introduces a mathematical potential analysis of PSO, revealing conditions causing stagnation and non-convergence even with good parameters, supported by experiments.

Findings

Potential in some dimensions decreases faster, causing loss of relevance.

Stagnation occurs frequently in polynomial and sphere functions.

Unmodified PSO often fails to converge to local optima under tested conditions.

Abstract

Particle Swarm Optimization (PSO) is a nature-inspired meta-heuristic for solving continuous optimization problems. In the literature, the potential of the particles of swarm has been used to show that slightly modified PSO guarantees convergence to local optima. Here we show that under specific circumstances the unmodified PSO, even with swarm parameters known (from the literature) to be good, almost surely does not yield convergence to a local optimum is provided. This undesirable phenomenon is called stagnation. For this purpose, the particles' potential in each dimension is analyzed mathematically. Additionally, some reasonable assumptions on the behavior if the particles' potential are made. Depending on the objective function and, interestingly, the number of particles, the potential in some dimensions may decrease much faster than in other dimensions. Therefore, these dimensions lose relevance, i.e., the contribution of their entries to the decisions about attractor updates becomes insignificant and, with positive probability, they never regain relevance. If Brownian Motion is assumed to be an approximation of the time-dependent drop of potential, practical, i.e., large values for this probability are calculated. Finally, on chosen multidimensional polynomials of degree two, experiments are provided showing that the required circumstances occur quite frequently. Furthermore, experiments are provided showing that even when the very simple sphere function is processed the described stagnation phenomenon occurs. Consequently, unmodified PSO does not converge to any local optimum of the chosen functions for tested parameter settings.