Particle Convergence Expected Time in the Stochastic Model of PSO
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 determini
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Abstract 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.
1 Introduction A predominance of one optimization method over another may depend on a difference either in the quality of the results for the given same computational cost or in the computational costs for the requested quality of the result. However, in the case of stochastic optimization algorithms, one can not guarantee that the optimal solution is found even in the finite amount of time. Therefore, an expected computational cost necessary to find a sufficiently good, suboptimal solution can be estimated at most. Additionally, due to typical weaknesses of these algorithms, like, for example, the tendency to get stuck in local optima, the expected cost cannot be estimated in general, but just for specific optimization environments. In spite of this, conclusions K. Trojanowski · T. Kulpa (B) Faculty of Mathematics and Natural Sciences, School of Exact Sciences, Cardinal Stefan Wyszy´nski University in Warsaw, Wóycickiego 1/3, 01-938 Warsaw, Poland e-mail: [email protected] K. Trojanowski e-mail: [email protected] © Springer Nature Switzerland AG 2019 J. J. Merelo et al. (eds.), Computational Intelligence, Studies in Computational Intelligence 792, https://doi.org/10.1007/978-3-319-99283-9_4
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K. Trojanowski and T. Kulpa
from such an analysis can be a source of improvements in real-world applications and thus convergence properties of these methods remain a subject of undiminished interest. A stochastic population-based optimization approach, precisely, a particle swarm optimization (PSO) is a subject of the presented research. In [1], authors indicate properties affecting the computational cost of finding suboptimal solutions, like particle stability, patterns of particle movements, or a local convergence of a particle and of a swarm. We are interested in the estimation of an expected runtime of the particle, precisely, the number of perturbations and respective fitness function calls necessary for the particle to obtain its stable state. For the stochastic model of the particle movement, new definitions of particle convergence based on the convergence of its expected location and expected variance of the locat
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