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

Abstract

Particle Swarm Optimization (PSO) is a nature-inspired meta-heuristic for solving continuous optimization problems. In [16,18], the potential of the particles of a 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 of 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.

High Precision Particle Swarm Optimization Algorithm (HiPPSO)

“…The relative potential captures the importance of each dimension and in [4] it has been argued that if this relative potential is quite low for some dimensions then the PSO does not optimize them.…”

“…Therefore it is obvious that the PSO actually optimizes only the last two dimensions. In [4] it has been argued that the relative potential in dimensions which are not optimized any longer decreases (on average) linearly in logarithmic scale. If only fixed precision is used, this insight would not be possible at all.…”

“…All the data has been produced by HiPPSO as it is possible to use fixed as well as selfadjusting precision for evaluation. The authors of [4] also used an early version of this implementation for their experiments.…”

“…The problem gets even worse if we look at further statistics obtained from the evaluations with two particles. In [4] a potential has been introduced to analyze bad convergence.…”

Performance evaluation of the particle swarm optimization algorithm to unambiguously estimate plasma parameters from incoherent scatter radar signals

“…This early convergence effect is known as a "stagnation problem" and it occurs when particles stop changing their positions, generating an early convergence to a position not guaranteed to be the global or local minimum (Freitas et al 2020). Additionally, it has been theoretically and experimentally demonstrated that the PSO algorithm would not converge to the optimum solution if the number of used particles is very small for the problem (Raß et al 2015).…”