Particle swarm stability: a theoretical extension using the non-stagnate distribution assumption

Abstract

This paper presents an extension of the state of the art theoretical model utilized for understanding the stability criteria of the particles in particle swarm optimization algorithms. Conditions for order-1 and order-2 stability are derived by modeling, in the simplest case, the expected value and variance of a particle's personal and neighborhood best positions as convergent sequences of random variables. Furthermore, the condition that the expected value and variance of a particle's personal and neighborhood best positions are convergent sequences is shown to be a necessary condition for order-1 and order-2 stability. The theoretical analysis presented is applicable to a large class of particle swarm optimization variants.

Boundary Constraint Handling Techniques for Particle Swarm Optimization in High Dimensional Problem Spaces

“…The experiments used PSO with inertia weight [22] with the global best topology. The selected inertia weight, w = 0.7298 and the acceleration coefficients c 1 = c 2 = 1.49618 are known good values suggested by Clerc [7] that guarantee convergence of the swarm (in terms of expectation and variance of particle positions [5]). Each swarm consisted of 50 particles.…”

The Importance of Component-Wise Stochasticity in Particle Swarm Optimization

“…The experiments used PSO with inertia weight as introduced in [17] with the global best topology. The selected inertia weight, w = 0.7298 and the acceleration coefficients c 1 = c 2 = 1.49618 are known good values suggested by Clerc [3] that guarantee convergence of the swarm (in terms of expectation and variance of particle positions [2]). As suggested by [1], all personal and global best positions were restricted to be within the search space.…”

Stability Analysis of the Multi-objective Multi-guided Particle Swarm Optimizer

“…To derive order-1 and order-2 stable regions for MGPSO, the following general theorem of Cleghorn and Engelbrecht [6] is used: Theorem 1. The following properties hold for all PSO variants of the form:…”

“…1 ρ(M) denotes the spectral radius of the matrix M Definition 1. Non-stagnant distribution assumption [6]: It is assumed thatŷ i (t), y i (t), andâ i (t) are random variables sampled from a time dependent distribution, such thatŷ i (t), y i (t), andâ i (t) have well defined expectations and variances for each t and that lim…”

“…Given that MGPSO has a similar structure to that of the original PSO [9] with the presence of inertia as proposed by Shi [16], existing PSO theory can be readily applied to the stability analysis of MGPSO. Specifically, Cleghorn and Engelbrecht [6] recently proposed a general theorem for deriving stability criteria for a class of PSO variants under minimal modeling assumptions. To the authors' knowledge this is the first paper to perform stability analysis of a multiobjective PSO.…”