Analytic solution of the continuous particle swarm optimization problem
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Analytic solution of the continuous particle swarm optimization problem Calogero Orlando1
· Angela Ricciardello1
Received: 28 January 2020 / Accepted: 11 November 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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% has always been obtained. Keywords Particle Swarm Optimization · Monte Carlo · Cauchy problem
1 Introduction Introduced by Eberhart and Kennedy in 1995, [1], the PSO is currently a heuristic optimization technique well established in the scientific community [2–5]. One of the drawback of the method is related to computational time requested to solve high dimensional problems and for such reason several modifications to the original scheme have been proposed. Among others a modification of PSO incorporating best human learning strategies has been proposed in [6] and the method has been compared with
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Calogero Orlando [email protected] Angela Ricciardello [email protected]
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University of Enna Kore, Enna, Italy
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C. Orlando , A. Ricciardello
others exiting PSO modification; in [7] a decline model for the swarm population has been proposed allowing to use a larger swarm during the initial exploration phases which is reduced during the exploitation phase obtaining a computational time reduction and preserving at the same time the global search capability of the swarm. A quantum behaved PSO has been proposed and its superiority has been verified in [8]. Eventually, in [9], major PSO-based algorithms and their applications have been analysed and compared. A continuous particle swarm optimization formulation (CPSO) is presented in this work with the aim of accelerating the convergence of the swarm toward the global best position. The particle position and velocity are updated by using continuous functions that are obtained by integrating a given Cauchy problem described by a second order ODE. The proposed algorithm is compared with standard PSO taking into account the minimization of a modified Rastrigin function. A Monte Carlo like approach is employed to account f
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