2014 IEEE International Conference on Systems, Man, and Cybernetics October 5-8, 2014, San Diego, CA, USA
Canonical Deterministic Particle Swarm Optimization to Sustain Global Search Kenya Jin’no, Takuya Shindo, Takuya Kurihara, Takefumi Hiraguri, Hideaki Yoshino Department of Electrical and Electronics Engineering Nippon Institute of Technology Saitama, Miyashiro 345–8501, Japan Email:
[email protected] Abstract—Particle swarm optimization is very simple algorithm. However, the trajectory of each particle becomes pretty complicated because each particle is affected by the information of the other particles. In order to analyze the dynamics of the PSO rigorously, we propose the canonical deterministic PSO (CD-PSO). The CD-PSO is described by a canonical form, and the dynamics is characterized by the damping factor and the rotational angle. The dynamics of the CD-PSO can be analyzed theoretically, however, the solution search performance is worse than the conventional PSO. The reason why the performance of the CD-PSO is poor, is the rotation radius of the particle is decreased with the change of the best position. The small rotation radius contributes to the local search ability, but the global search ability is spoiled. Therefore, we propose a method to sustain the global search for the CD-PSO. The function is realized by maintaining the rotation radius. Applying the proposed method for the CD-PSO, the solution search performance improves that of the conventional CD-PSO.
I.
I NTRODUCTION
Particle swarm optimization (PSO), which was originally proposed by J. Kennedy and R. Eberhart [1][2], is developed as a global optimal solution searching algorithm. The PSO is classified into one of swarm intelligence algorithms. The PSO works with a set of particles where their positions denote potential solutions. The moving direction of each particle is influenced by its own best position and the neighbor’s best information of knowable best solutions. This simple fundamental concept exhibits remarkable search performance, and the implementation of this algorithm is easy. Therefore, the PSO algorithm have been applied to various optimization problems [3][4][5][6]. Also many algorithms to improve the search performance have been proposed [7]. The particles of the conventional PSO are operated by the simple rule. However, the trajectory of each particle becomes pretty complicated because each particle is affected by the information of the other particles. Therefore, the theoretical analysis of the dynamics is hard. In order to analyze the dynamics rigorously, the deterministic PSO have been proposed and have been analyzed[8][9]. This deterministic PSO is proposed to analyze its dynamics, so its solution search performance is degraded comparing with the conventional PSO. However, the dynamics is described by simple equations whose algorithm is easy to understand. We have analyzed the dynamics of the deterministic PSO (D-PSO) whose parameters are converted to a damping factor and a rotation angle [10]. Based on the analyzed results of the D-PSO, we have proposed a canonical deterministic PSO (CD-PSO) [11] In this article,
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we propose a method that the rotation radius in the phase space is kept to improve the search performance of the CD-PSO. Each particle of the conventional PSO memorizes the position information that gives the best evaluation value. The best position information is named personal best position pbesttj . The particles also share the best position information in the swarm. The best position in the swam is named global best position gbestt . The position of each particle and its velocity is calculated as the followings: $ % ⎧ t+1 t t t − x ⎨ vj = wvj + c1 r1 pbest j j $ % +c2 r2 gbestt − xtj (1) ⎩ t+1 xj = xtj + vjt+1
where xtj is the position vector of the j-th particle on the tth iteration, vjt is the velocity vector of the j-th particle on the t-th iteration, w is an inertia parameter, and c1 and c2 are acceleration parameters. r1 and r2 are uniformly distribute function whose range is [0, 1]. II.
D ETERMINISTIC PSO
The conventional PSO which is described by Eq. (1), includes stochastic elements, namely, uniform distributed random numbers; r1 and r2 . Therefore, the conventional PSO is regarded as a stochastic system. If these parameters are regarded as a constant that is r1 = r2 = 1, the system of Eq. (1) can be regarded as a deterministic system. Such system is said to be deterministic PSO (D-PSO). Here, the best position information is recast into the following expression. & t pj = γpbesttj + (1 − γ)gbestt (2) 1 γ = c1c+c 2 where ptj denotes an origin of the rotation of each particle in the phase space. In this case, Eq. (1) is transformed into the following matrix form: ' t+1 ( ' (' t ( vj vj w −c = (3) w 1−c yjt yjt+1 where yjt = xtj − ptj .
Note that this system does not contain stochastic factor, therefore, this system can be regarded as a deterministic system.
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The best position information ptj is equivalent to the global best if γ = 0 is satisfied. On the other hand, if γ = 1 is satisfied, ptj is equivalent to the personal best. Since each dimension of the position of the particle is independent, without loss of generality, we can consider onedimensional case to analyze the dynamics. One-dimensional D-PSO is described by ' t+1 ( ' (' t ( vj vj w −c = . (4) w 1−c yjt yjt+1 The behavior of the system is governed by eigenvalues of the matrix in Eq. (4). The eigenvalues λ are calculated as ) (1 + w − c) ± (1 + w − c)2 − 4w . (5) λ= 2 Since the D-PSO is a discrete time system, the system is stable iff the eigenvalues λ exist within the unit circle on the complex plane. If the parameters satify (1 + w − c)2 − 4w ≥ 0,
the eigenvalues λ must be real numbers. In this case, if & |w| < 1 0 < c < 2w + 2
(6)
(7)
are satisfied, the eigenvalues exist within the unit circle on the complex plane. If the parameters satisfy (1 + w − c)2 + 4w < 0,
(8)
the eigenvalues λ must be complex eigenvalues. In this case, if 0
