Abstract
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.
Highlights
Particle swarm optimization (PSO) is an evolutionary computation technique [1] for optimization which is inspired by the social behavior of individuals in groups in nature
We present a generalized form of the particle swarm optimization (PSO), which is derived from a continuous version of PSO
(1) The deterministic and stochastic stability regions and the asymptotic velocities of convergence of the Generalized PSO (GPSO) algorithm are analyzed as a function of the time step, the local and global accelerations, and the inertia value
Summary
Particle swarm optimization (PSO) is an evolutionary computation technique [1] for optimization which is inspired by the social behavior of individuals in groups in nature. Convergence properties and PSO trajectories were analyzed, and helped to clarify some numerical aspects of the PSO algorithm; see [2,3,4,5,6] Based on these analysis, some parameter selection strategies were proposed (see, e.g., [5]). (1) The deterministic and stochastic stability regions and the asymptotic velocities of convergence of the GPSO algorithm are analyzed as a function of the time step, the local and global accelerations, and the inertia value. (4) a comparison is made between PSO and the GPSO by means of numerical experiments using wellknown benchmark functions supporting the theoretical results Based on these results two variants of generalized PSO algorithm are proposed, improving the convergence and the exploration task while solving real applications in inverse problems modeling
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