Theoretical analysis of particle swarm trajectories through a mechanical analogy

Fernández-Martínez, Juan Luis; García–Gonzalo, Esperanza; Fernández-Álvarez, José Paulino

doi:10.5019/j.ijcir.2008.129

Cited by 50 publications

(26 citation statements)

References 9 publications

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“…that provides the continuous analytical function that overlaps the discrete PSO points, and allows us to perform a systematic study of the PSO trajectories under stagnation [9]. Four different zones have been differentiated (see Figure 1) showing a different behavior, depending on the character of the eigenvalues of the characteristic equation (8) associated to the PSO difference equation.…”

Section: The Pso Difference Equationsmentioning

confidence: 99%

“…The deterministic stability region with moving center of attraction turns to be S D . Also, this methodology allows us to obtain information about the asymptotic velocity of convergence, v c (w, φ), as a function the spectral radius ρ of the iteration matrix M PSO [9]:…”

Section: The Pso Difference Equationsmentioning

confidence: 99%

“…The PSO differential model (15) has been deduced from a mechanical analogy [9]: a damped mass-spring system of unit mass, damping factor, b = 1 − w, and stiffness constant, k = φ. The knowledge of the trajectories of both the continuous and the discrete model helped to explain why some zones of the parameter space (w, φ) provide better convergence, in terms of some combined criteria: trajectory attenuation, trajectory oscillation, and center attraction potential.…”

Section: The Pso Continuous Modelmentioning

confidence: 99%

“…Fernández Martínez et al [9] mentioned this effect, comparing the PSO stability region to its continuous counterpart. Border line (32) separating real and complex roots also follows a similar behavior.…”

Section: Stagnation Casementioning

confidence: 99%

“…Physical analogy with a damped mass-spring oscillator was introduced by Brandstätter and Baumgartner [8] to optimize electrical engineering problems, nevertheless, this analogy was not completely exploited in their work. Recently, Fernández Martínez et al [9] used this analogy to derive the PSO continuous model and to present a systematic study of PSO trajectories by means of stability analysis of the deterministic PSO difference equations involved and fixed point techniques. The PSO convergence can then be explained in terms of some combined criteria: trajectory attenuation, trajectory oscillation, and center attraction potential, explaining the success in achieving convergence of some parameter sets found in the literature.…”

Section: Introductionmentioning

confidence: 99%

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The Generalized PSO: A New Door to PSOEvolution

Laviada

García–Gonzalo

2008

Journal of Artificial Evolution and Applications

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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 illposedness 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.

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Section: The Pso Difference Equationsmentioning

confidence: 99%

Section: The Pso Difference Equationsmentioning

confidence: 99%

Section: The Pso Continuous Modelmentioning

confidence: 99%

Section: Stagnation Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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