Analytic solution of the continuous particle swarm optimization problem

The discrete formulation of Particle Swarm Optimization (PSO) is nowadays widely used. The paper presents a continuous formulation of the PSO problem along with its analytic solution. The aim is to verify whenever an amelioration of the standard discrete PSO is achievable by employing its continuous counterpart. The convergence of the proposed continuous PSO scheme is analyzed accounting for variation of the algorithm’s parameters. Moreover, looking for the minimization of an a-priori chosen modified Rastringrin function, a comparison with the standard PSO is also given in terms of computational time and likelihood of success of finding the global optimum points using a Monte Carlo like analysis to consider the stochastic nature of the PSO. Last, comparisons with other optimization methods such as genetic algorithm and tabu search as well as with some extension PSO methods have been investigated. Different objective functions have been taken into account and a success rate greater that 93%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$93\%$$\end{document} has always been obtained.