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

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