Particle Convergence Time in the Deterministic Model of PSO
A property of particles in Particle Swarm Optimization (PSO), namely, particle convergence time (pct) is a subject of theoretical and experimental analysis. For the model of PSO with inertia weight a new measure for evaluation of pct is proposed. The meas
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Abstract. A property of particles in Particle Swarm Optimization (PSO), namely, particle convergence time (pct) is a subject of theoretical and experimental analysis. For the model of PSO with inertia weight a new measure for evaluation of pct is proposed. The measure evaluates number of steps necessary for a particle to obtain a stable state defined with any precision. For this measure an upper bound formula of pct is derived and its properties are studied. Four main types of particle behaviour characteristics are selected and discussed. In the experimental part of the research effectiveness of swarms with different characteristics of their members are verified. A new type of swarm control improving efficiency of a swarm in escaping traps of local optima is proposed and experimentally verified.
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Introduction
Particle swarm optimization (PSO) [1] belongs to a big family of modern heuristic optimization methods. A number of versions of PSO has already been proposed sharing the same paradigm of stochastic, population-based method of exploration in the given space of solutions in searching for the best one. In our research we selected one of the earlier versions of PSO proposed in [2]. Like in other methods, the population consists of members called here particles which represent solutions from the given space. Particles are also equipped with memories which store attractors, that is, solutions best found so far by the particles. A working group of particles controlled by the method is called a swarm. After the initialization of a swarm the cycle of iterations performs the search process. The distinctive features of PSO are: (1) application of particle memory as well as the mechanism of memory sharing by groups of neighbouring solutions, (2) the method of finding new solutions based on the idea of displacement originated from the real-world. Unlike other metaheuristics, every iteration consists of two main steps: particles memory update and the displacement of particles within the space of solutions. In PSO less-fit particles do not die, that is, there is no “survival of the fittest” mechanism typical for the evolutionary approach. The rules of displacement make use of the information from the memory and are expressed by equations which may differ to each other for different versions c Springer International Publishing AG 2017 J.J. Merelo et al. (eds.), Computational Intelligence, Studies in Computational Intelligence 669, DOI 10.1007/978-3-319-48506-5 10
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K. Trojanowski and T. Kulpa
of PSO. Particularly, in the version of PSO which we selected for analysis the rules of displacement use the inertia weight parameter. Numerous applications of PSO confirmed its usefulness and potential but also motivate for studying their theoretical properties. Particularly, a particle stability analysis is a subject of great interest. One of the main aims is estimation of particle parameter ranges guaranteing the convergent movement within the given boundaries of the search space. For the purpose of theoretical analysis some assumptions
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