Particle swarm optimization almost surely finds local optima

Abstract

Particle swarm optimization (PSO) is a popular natureinspired meta-heuristic for solving continuous optimization problems. Although this technique is widely used, up to now only some partial aspects of the method have been formally investigated. In particular, while it is well-studied how to let the swarm converge to a single point in the search space, no general theoretical statements about this point or on the best position any particle has found have been known. For a very general class of objective functions, we provide for the first time results about the quality of the solution found. We show that a slightly adapted PSO almost surely finds a local optimum by investigating the newly defined potential of the swarm. The potential drops when the swarm approaches the point of convergence, but increases if the swarm remains close to a point that is not a local optimum, meaning that the swarm charges potential and continues its movement.

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

“…No bound handling strategies are investigated, because they have almost no influence on the convergence if the swarm is converging to a point not on the boundaries. Similar to [16,18], the model describes the positions of the particles, the velocities and the global and local attractors also as real-valued stochastic processes. Basic mathematical tools from probability theory, which are needed for this analysis can be found in, e. g., [5].…”

“…Now the idea of a stagnation measure is introduced, which is a multidimensional extension to the potential used in [16,17,18]. For every step, a D-dimensional vector of potentials is evaluatedone potential value for each dimension.…”

“…This results in a running phase in that dimension. Running phases are introduced in [16,17,18]. If the swarm is running in a dimension d, then the coordinate value of dimension d determines the local attractor and the influence of the other dimensions can be neglected.…”

“…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.…”

“…In [16,18], the notion of the potential of the swarm has been introduced. This potential has been used to prove that in the one-dimensional search space the swarm almost surely (in the mathematical sense) finds a local optimum.…”