Convergence and Mathematical Analysis of Particle Swarm Optimization

Abstract

The particle swarm optimization(PSO) is one of a very useful heuristic algorithms. In PSO, we investigate the simulation of natural swarm behaviors, such as those of birds or insects, which search for significant areas by exchanging information and cooperating rather than competing. A number of artificial particles move through the domain RD, with each particle’s movement influenced by its own experiences as well as those of the swarm agents. In particular, for guar anteeing convergence, necessary and sufficient conditions to a certain swarm parameters that control the behavior of the swarm could be derived. In order to measure, how fast the swarm at a certain time converges, we define and analyze the potential of parameters on a particle swarm. When the swarm is far away from a local optimum, the potential increases in the 1 dimensional case. This reflection turns out to be sufficient to prove the main results, namely that in a 1-dimensional situation, the swarm with probability 1 converges towards a local opti mum for a comparatively wide range of objective functions. Additionally, we applied a Markov chain to demonstrate that, for benchmark function tests, the numerical simulation of the PSO algorithm convergence analysis shows that the constriction standard particle swarm optimiza tion (CSPSO) algorithm enhances iteration generation performance. In the general dimension the swarm might not converge towards a local optimum. Instead, it gets adhered to a position where some dimensions have a potential orders of size smaller than others. In such situations the algorithm converges towards a point in the search space, is not even a local optimum which is called swarm stagnates. In order to solve this issue, we propose a variant of PSO, such as standard particle swarm optimization repulsive functional constraint (SPSO-RFC) velocity to zero that guarantees convergence towards a local optimum. As a result, the CSPSO and PSO RFC algorithms demonstrate better optimal solutions compared to the PSO variants, owing to the effects of parameter settings within the PSO algorithm.