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Particle Convergence Expected Time in The PSO Model with Inertia Weight
Abstract
Theoretical properties of particle swarm optimization approach with inertia weight are investigated. Particularly, we focus on the convergence analysis of the expected value of the particle location and the variance of the location. Four new measures of the expected particle convergence time are defined: (1) convergence of the expected location of the particle, (2) the particle location variance convergence and (3-4) their respective weak versions. For the first measure an explicit formula of its upper bound is also given. For the weak versions of the measures graphs of recorded values are presented.
... For the deterministic model, a particle convergence time (pct) was proposed in [16]. For the stochastic model, in [17,18] authors define two measures: a convergence expected time (pcet) and a particle location variance convergence time (pvct). The pcet(δ) measure represents the minimal number of steps necessary for the expected particle location to attain equilibrium with the given precision δ, and the pvct(δ) measure -a minimal number of steps necessary to get variance of particle location lower than δ for all subsequent time steps. ...
... The measures are hardly applied in practice, thus, authors propose also weak versions of the two measures where the first expected difference of particle location or the first variance of particle location lower than δ are the stopping conditions in the measurement process. Characteristics of weak measures for different particle configurations can be found in [17]. The characteristics were generated for the stochastic model of PSO with inertia weight under stagnation assumption. ...
... The measures of pcet(δ) and pvct(δ) proposed in [17] evaluate time necessary for a particle to reach an equilibrium state. Due to the individual evaluation of velocity vector coordinates in IPSO, the definitions of the two measures concern a one-dimensional search space: Definition 1 (The particle convergence expected time:). ...
Stability properties of a particle in the stochastic models of PSO are a subject of presented analysis. Measures of a number of particle steps necessary to reach an equilibrium state, particularly, generalized weak versions of measures: particle convergence expected time (pcet) and the particle location variance convergence time (pvct) are developed. A new measure, namely particle stability time, is also proposed. For all the measures graphs of estimated and recorded values are presented. Finally, an adaptation of expected running time (ERT) measure is proposed which can be applied for identification of convergence regions in the parameters space.
In Particle Swarm Optimization, the behavior of particles depends on the parameters of movement formulas. In our research, we identify types of particles based on their movement trajectories. Then, we propose new rules of particle classification based on the two attributes of the measure representing the minimum number of steps necessary for the expected particle location to obtain its stable state. The new classification clarifies the division into types of particles based on the observation of different shapes of their movement trajectories.
Convergence properties in the model of PSO with inertia weight are a subject of analysis. Particularly, we are interested in estimating the time necessary for a particle to obtain equilibrium state in deterministic and stochastic models. For the deterministic model, an earlier defined upper bound of particle convergence time (pctb) is revised and updated. For the stochastic model, four new measures of the expected particle convergence time are proposed: (1) the convergence of the expected location of the particle, (2) the particle location variance convergence and (3)–(4) their respective weak versions. In the experimental part of the research, graphs of recorded expected running time (ERT) values are compared to graphs of upper bound of pct from the deterministic model as well as graphs of recorded convergence times of the particle location pwcet from the stochastic model.
Particle Swarm Optimization (PSO) is a powerful heuristic optimization method being subject of continuous interest. Theoretical analysis of its properties concerns primarily the conditions necessary for guaranteeing its convergent behaviour. Particle behaviour depends on three groups of parameters: values of factors in a velocity update rule, initial localization and velocity and fitness landscape. The paper presents theoretical analysis of the particle convergence properties in the model with inertia weight respectively to different values of these parameters. A new measure for evaluation of a particle convergence time is proposed. For this measure an upper bound formula is derived and its four main types of characteristics are discussed. The way of the characteristics transformations respectively to changes of velocity equation parameters is presented as well.
This paper presents an objective function specially designed for the convergence analysis of a number of particle swarm optimization (PSO) variants. It was found that using a specially designed objective function for convergence analysis is both a simple and valid method for performing assumption free convergence analysis. It was also found that the canonical particle swarm’s topology did not have an impact on the parameter region needed to ensure convergence. The parameter region needed to ensure convergent particle behavior was empirically obtained for the fully informed PSO, the bare bones PSO, and the standard PSO 2011 algorithm. In the case of the bare bones PSO and the standard PSO 2011, the region needed to ensure convergent particle behavior differs from previous theoretical work. The difference in the obtained regions in the bare bones PSO is a direct result of the previous theoretical work relying on simplifying assumptions, specifically the stagnation assumption. A number of possible causes for the discrepancy in the obtained convergent region for the standard PSO 2011 are given.
A number of theoretical studies of particle swarm optimization (PSO) have been done to gain a better understanding of the dynamics of the algorithm and the behavior of the particles under different conditions. These theoretical analyses have been performed for both the deterministic PSO model and more recently for the stochastic model. However, all current theoretical analyses of the PSO algorithm were based on the stagnation assumption, in some form or another. The analysis done under the stagnation assumption is one where the personal best and neighborhood best positions are assumed to be non-changing. While analysis under the stagnation assumption is very informative, it could never provide a complete description of a PSO’s behavior. Furthermore, the assumption implicitly removes the notion of a social network structure from the analysis. This paper presents a generalization to the theoretical deterministic PSO model. Under the generalized model, conditions for particle convergence to a point are derived. The model used in this paper greatly weakens the stagnation assumption, by instead assuming that each particle’s personal best and neighborhood best can occupy an arbitrarily large number of unique positions. It was found that the conditions derived in previous theoretical deterministic PSO research could be obtained as a specialization of the new generalized model proposed. Empirical results are presented to support the theoretical findings.
Abstract Several stability analyses and stable regions of particle swarm optimization (PSO) have been proposed before. The assumption of stagnation and different definitions of stability are adopted in these analyses. In this paper, the order-2 stability of PSO are analyzed based on a weak stagnation assumption. A new definition of stability is proposed and an order-2 stable region is obtained. Several existing stable analyses for canonical PSO are compared, especially their definitions of stability and the corresponding stable regions. It is shown that, the classical stagnation assumption is too strict and not necessary. Moreover, among all these definitions of stability, it is shown that our definition requires the weakest conditions, and additional conditions bring no benefit. Finally, numerical experiments are reported to show that the obtained stable region is meaningful. A new parameter combination of PSO is also shown to be good, even better than some known best parameter combinations.
Particle swarm optimisation (PSO) has been enormously successful. Within little more than a decade hundreds of papers have reported successful applications of PSO. In fact, there are so many of them, that it is difficult for PSO practitioners and researchers to have a clear up-to-date vision of what has been done in the area of PSO applications. This brief paper attempts to fill this gap, by categorising a large number of publications dealing with PSO applications stored in the IEEE Xplore database at the time of writing.
Several theoretical analyses of the dynamics of particle swarms have been offered in the literature over the last decade. Virtually all rely on substantial simplifications, often including the assumption that the particles are deterministic. This has prevented the exact characterization of the sampling distribution of the particle swarm optimizer (PSO). In this paper we introduce a novel method that allows us to exactly determine all the characteristics of a PSO sampling distribution and explain how it changes over any number of generations, in the presence stochasticity. The only assumption we make is stagnation, i.e., we study the sampling distribution produced by particles in search for a better personal best. We apply the analysis to the PSO with inertia weight, but the analysis is also valid for the PSO with constriction and other forms of PSO.
The particle swarm optimization algorithm is analyzed using standard results from the dynamic system theory. Graphical parameter selection guidelines are derived. The exploration-exploitation tradeoff is discussed and illustrated. Examples of performance on benchmark functions superior to previously published results are given.
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This paper reviews recent studies on Particle Swarm Optimization (PSO) algorithm. The review has been focused on high impact recent articles that have analyzed and/or modified PSO algorithms. This paper also presents some potential areas for future study.
This letter presents a formal stochastic convergence analysis of the standard particle swarm optimization (PSO) algorithm, which involves with randomness. By regarding each particle's position on each evolutionary step as a stochastic vector, the standard PSO algorithm determined by non-negative real parameter tuple {omega, c(1), c(2)) is analyzed using stochastic process theory. The stochastic convergent condition of the particle swarm system and corresponding parameter selection guidelines are derived. (c) 2006 Elsevier B.V. All rights reserved.