Table of Contents
Journal of Artificial Evolution and Applications
Volume 2008, Article ID 861275, 15 pages
Research Article

The Generalized PSO: A New Door to PSO Evolution

Department of Mathematics, University of Oviedo, C/Calvo Sotelo S/N, 33007 Oviedo, Spain

Received 20 July 2007; Revised 15 January 2008; Accepted 24 February 2008

Academic Editor: Riccardo Poli

Copyright © 2008 J. L. Fernández Martínez and E. García Gonzalo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A generalized form of the particle swarm optimization (PSO) algorithm is presented. Generalized PSO (GPSO) is derived from a continuous version of PSO adopting a time step different than the unit. Generalized continuous particle swarm optimizations are compared in terms of attenuation and oscillation. The deterministic and stochastic stability regions and their respective asymptotic velocities of convergence are analyzed as a function of the time step and the GPSO parameters. The sampling distribution of the GPSO algorithm helps to study the effect of stochasticity on the stability of trajectories. The stability regions for the second-, third-, and fourth-order moments depend on inertia, local, and global accelerations and the time step and are inside of the deterministic stability region for the same time step. We prove that stability regions are the same under stagnation and with a moving center of attraction. Properties of the second-order moments variance and covariance serve to propose some promising parameter sets. High variance and temporal uncorrelation improve the exploration task while solving ill-posed inverse problems. Finally, a comparison is made between PSO and GPSO by means of numerical experiments using well-known benchmark functions with two types of ill-posedness commonly found in inverse problems: the Rosenbrock and the “elongated” DeJong functions (global minimum located in a very flat area), and the Griewank function (global minimum surrounded by multiple minima). Numerical simulations support the results provided by theoretical analysis. Based on these results, two variants of Generalized PSO algorithm are proposed, improving the convergence and the exploration task while solving real applications of inverse problems.