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