The Generalized PSO: A New Door to PSOEvolution

A generalized form of the particle swarm optimization (PSO) algorithm ispresented. Generalized PSO (GPSO) is derived from a continuous version of PSO adopting atime step different than the unit. Generalized continuous particle swarm optimizations are compared in terms ofattenuation and oscillation. The deterministic and stochastic stability regions and their respectiveasymptotic velocities of convergence are analyzed as a function of the time step and theGPSO parameters. The sampling distribution of the GPSO algorithm helps to study the effectof stochasticity on the stability of trajectories. The stability regions for the second-, third-, andfourth-order moments depend on inertia, local, and global accelerations and the time step and areinside of the deterministic stability region for the same time step. We prove that stability regionsare 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. Highvariance and temporal uncorrelation improve the exploration task while solving ill-posed inverseproblems. Finally, a comparison is made between PSO and GPSO by means of numerical experimentsusing well-known benchmark functions with two types of ill-posedness commonly found in inverseproblems: the Rosenbrock and the “elongated” DeJong functions (global minimum located in avery 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 theexploration task while solving real applications of inverse problems.