Abstract
AbstractThe existing stability analysis of particle swarm optimization (PSO) algorithm is chiefly concluded by the assumption of constant transfer matrix or time-varying random transfer matrix. Firstly, one counterexample is provided to show that the existing convergence analysis is not possibly valid for PSO system involving random variables. Secondly, the joint spectral radius, mainly calculated by the maximum eigenvalue of the product of all asymmetric random transfer matrices, is introduced to analyze and discuss convergence condition and convergence rate from numerical viewpoint with the aid of Monte Carlo method. Numerical results show that there is one major discrepancy between some preview convergence results and our corresponding results, helping us to deeply understand the tradeoff between exploration ability and exploitation ability as well as providing certain guideline for parameter selection.
Highlights
Particle swarm optimization (PSO), firstly developed by Kennedy and Eberhart in 1995 [1][2], is essentially one typical stochastic global optimization method
From the perspective of the joint spectral radius, we are mainly concerned with analyzing the stability and discuss parameter selection of the standard PSO algorithm
One new convergence analysis on the random PSO algorithm is provided to discuss the difference between the existing results and our results
Summary
Particle swarm optimization (PSO), firstly developed by Kennedy and Eberhart in 1995 [1][2], is essentially one typical stochastic global optimization method. 2. The joint spectral radius, which is firstly introduced and defined in this paper to discuss the stability of the standard PSO algorithm, measures convergence rate of all particles and describes the mathematical tradeoff between exploration ability and exploitation ability. 3. Because of asymmetric transfer matrix involving random variables in PSO system, it is very challenging to calculate the final product matrix and the joint spectral radius, which determines the stability of the standard PSO algorithm and the convergence rate. Because of asymmetric transfer matrix involving random variables in PSO system, it is very challenging to calculate the final product matrix and the joint spectral radius, which determines the stability of the standard PSO algorithm and the convergence rate To handle with this problem, Monte Carlo method is utilized to analyze convergence condition and discuss parameter selection.
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