STOCHASTIC STABILITY AND NUMERICAL ANALYSIS OF TWO NOVEL ALGORITHMS OF THE PSO FAMILY: PP-GPSO AND RR-GPSO

摘要

The PSO algorithm can be physically interpreted as a stochastic damped mass-spring system: the so-called PSO continuous model. Furthermore, PSO corresponds to a particular discretization of the PSO continuous model. Based on this mechanical analogy we derived in the past a family of PSO-like versions, where the acceleration is discretized using a centered scheme and the velocity of the particles can be regressive (GPSO), progressive (CP-GPSO) or centered (CC-GPSO). Although the first and second order trajectories of these algorithms are isomorphic, CC-GPSO and CP-GPSO are very different from GPSO. In this paper we present two other PSO-like methods: PP-GPSO and RR-GPSO. These algorithms correspond respectively to progressive and regressive discretizations in acceleration and velocity. PP-PSO has the same velocity update than GPSO, but the velocities used to update the trajectories are delayed one iteration, thus, PP-PSO acts as a Jacobi system updating positions and velocities at the same time. RR-GPSO is similar to a GPSO with stochastic constriction factor. Both versions have a very different behavior from GPSO and the other family members introduced in the past: CC-PSO and CP-PSO. RR-PSO seems to have the greatest convergence rate and its good parameter sets can be calculated analytically since they are along a straight line located in the first order stability region. Conversely PP-PSO seems to be a more explorative version, although the behavior of these algorithms can be partly problem dependent. Both exhibit a very peculiar behavior, very different from other family members, and thus they can be called distant PSO relatives. RR-PSO have the greatest convergence rate of all family members for a wide range of benchmark functions with different numerical complexity in 10, 30 and 50 dimensions. These algorithms have been successfully applied for protein secondary structure prediction and in oil and gas reservoir optimization.