2011 International Workshop on Nonlinear Circuits,Communication and Signal Processing
NCSP'11, Tianjin, China, March 1-3, 2011

Particle Swarm Optimization
Containing Plural Swarms Using Shared Velocity†Masaki SUGIMOTO, ‡Haruna MATSUSHITA and †Yoshifumi NISHIO†Tokushima University
2-1 Minami-Josanjima,
Tokushima, Japan
Phone:+81-88-656-7470
FAX:+81-88-656-7471
Email: { sugimoto, nishio } @ee.tokushima-u.ac.jp

Abstract‡Kagawa University
2217-20 Hayashi-cho, Takamatsu,
Kagawa, Japan
Phone:+81-87-864-2237
FAX:+81-87-864-2262
Email:haruna@eng.kagawa-u.ac.jp

PSOV is that each particle of PSOV shares the same velocity(called shared velocity) that represented the average velocity
Particle Swarm Optimization (PSO) is a popular optimization of all the particles. Therefore, the velocity of each particle
technique for solving objective functions and an algorithm to influence the other particle velocity, each other. The study
simulate the movement of flocks of birds toward foods. In this showed that the shared velocity is eﬀective for the optimizastudy, we propose an improved PSO algorithm; PSO contain- tions using PSO.
ing shared velocity using plural swarms (PPSOV). Each partiIn this study, we propose an improved PSO algorithm; PSO
cle of PPSOV belongs to one of plural swarms. Each particle containing plural swarms using shared velocity (PPSOV).
belonging to each swarm shares a velocity with other parti- The important features of PPSOV are follows. PPSOV concles in same swarm. We evaluate the performance of PPSOV sists of plural swarms. The number of particles of each swarm
by using four optimization functions and confirm that PPSOV is all the same.
obtain the best results among four PSOs.
A mean velocity called “shared velocity” is calculate forevery swarm, and the each particle shares the shared velocity
with all the particles in the swarm. Thus, each swarm has the1. Introduction
diﬀerent shared velocity from other swarm. At every generation, the shared velocity of each swarm is updated depending
Particle Swarm Optimization (PSO) [1] is a popular option the velocity of the particles belonging to each swarm.
mization technique for solving objective functions and is alWe explain the algorithm of PPSOV in detail in Section 2.
gorithm to simulate the movement of flocks of birds toward
In Section 3, we perform basic numerical experiments by usfoods. Due to its simple concept, easy implementation anding four algorithm methods of PPSOV and other PSOs. Fromquick convergence, PSO has attracted attentions and has beenthese results, we confirm that the mean values of PPSOV are
widely applied to diﬀerent fields in recent years. Furtherthe best among four PSOs, therefore, we can confirm thatmore, PSO has demonstrated great performances for manyPPSOV algorithm is more eﬀective than the standard PSO.
optimization problems.
However, quick convergence often leads to a local optimum problem. It is important for multimodal functions with 2. PPSOV
a lot of local optima to compromise between the quick conThe standard PSO consists of M particles. Each particles
vergence and being trapped in a local optimum. In order to escape from such local optima and to avoid the premature con- has two information; position and velocity. The position vecvergence, the search for a global optimum should be diverse. tor of each particle i and its velocity vector are represented
Many researchers have improved the performance of PSO by by Xi = (xi1 , xi2 , . . . , xiD ) and V i = (vi1 , vi2 , . . . , viD ), respecenhancing its ability with more diverse search [2][3]. In par- tively, where (i = 1, 2, . . . , M) and xid ∈ [xmin , xmax ].
ticular, some researchers have proposed the PSO method usIn this study, we propose the PSO containing plural swarms
ing plural swarms whose particles exchange their information using shared velocity (PPSOV). PPSOV consists of not only
among particles belong each swarm [4][5]. In real world, a uni-swarm but plural swarms. The number of swarms is K,
bird becomes a flock that matched the speed and flies.
and each swarm is denoted by S k (k = 1, 2, · · · , K). Each parFrom this idea, we studied the improved PSO algorithm that ticle of PPSOV is sorted into one of swarms, and the number
using shared velocity (PSOV) [6]. The important feature of of particles in one swarm is M/K. The particles belonging- 203 -

8084000

6300042000

21000024020022060−200−2(a)500−2−20502−5(b)

−5(c)
2.5

200

2150

1.5

100

150

0.50500

010500

0−500

10000

−10−500(d)−10(e)

Figure 1: Landscape of four bench marks for D = 2. (a) Sphere function. (b) Rosenbrock’s function. (c) Rastrigin’s function.
(d) Griewank’s function with full definition range. (e) Inner area of Griewank’s function.

to each swarm share same velocity (called “shared velocity”)
with other particles in the swarm.
[PPSOV1] (Initialization) Let a generation step t = 0. Randomly initialize the position Xi and its velocity V i for all particles i and initialize Pi with a copy of Xi .
[PPSOV2] Evaluate the current cost f (Xi ). Update the personal best position pbest Pi = (pi1 , pi2 , ..., piD ) for each particle i and the global best position gbest Pg = (pg1 , pg2 , ..., pgD )among all the particles.
[PPSOV3] Let Psk = (p sk 1 , p sk 2 , · · · , p sk D ) represents theswarm best position with the best cost among the particles
belonging to the swarm S k so far (called sbest). Update Psk
for each swarm S k if needed.
sk = arg min{ f (Xi )},ii ∈ S k.

(1)

[PPSOV4] Updated V i and Xi of each particle i depending
on its pbest, its swarm best sbest and gbest;vid (t + 1) =wk vid + c1 r1 {pid − xid (t)}{}+ c2 r2 p sk d − xid (t) + c3 r3 {pgd − xid (t)},(2)where r1 , r2 and r3 are three random variables distributed uniformly on [0, 1], w is an inertia weight of all particles, and c1 ,c2 and c3 are positive acceleration coeﬃcients.
[PPSOV5] All the particles in each swarm S k shares theshared velocity V sk = (v sk1 , v sk2 , ..., v skD ). V sk is defined as
the average velocity of all the particles belonging to S k ;1 ∑vid ,M/K i=1
M/Kv sk d =i ∈ S k.

(3)

[PPSOV6] Update Xi depending on its V s and V i ;xid (t + 1) = xid (t) + cv v sk d (t + 1) + vid (t + 1),(4)

where cv is the shared velocity coeﬃcients in each swarm S k .
[PPSOV7] Let t = t + 1. Go back to [PPSOV2], and repeat
until t = T .
3. Numerical Experiments
In order to confirm the performance of PPSOV algorithm,
we carry out basic numerical experiments. The goal is tofind the optimum (minimum) value of f (x). Referring to [2],we use the following four benchmark functions as shown inFig. 1.

- 204 -Table 1: Comparison results PSO, PPSO, PSOV and PPSOV on test functions with D = 60.
ff1

f2

f3

f4

Avg.
Min.
Max.
Avg.
Min.
Max.
Avg.
Min.
Max.
Avg.
Min.
Max.

PSO
15.73
3.69E-06
32.77
286.10154.16554.93275.52
157.17380.3281.79
22.99
181.00

PPSO
12.49
1.77
55.43
313.17206.45498.25361.62
197.06686.85
11.84
3.71E-02
73.31

1. Sphere function:
f1 (x) =D∑xd2 ,(5)d=1where x ∈ [−2.048, 2.047]D and the optimum solution x∗ are
all [0, 0, . . . , 0].
2. Rosenbrock’s function:
f2 (x) =D−1∑
(100(xd2 − xd+1 )2 + (1 − xd )2 ),

(6)

d=1

where x ∈ [−2.048, 2.047]D and the optimum solution x∗ are
all [1, 1, . . . , 1].
3. Rastrigin’s function:
f3 (x) =D∑(xd2 − 5 cos(2πxd ) + 5),

(7)

d=1

where x ∈ [−5.12, 5.12]D and the optimum solution x∗ are all[0, 0, . . . , 0].
4. Griewank’s function:
f4 (x) =DD∑∏xd2
xdcos( √ ) + 1,+4000
dd=1
d=1(8)

PSOV
9.83
2.43E-16
19.66
133.0053.71
228.46267.81
196.28396.8356.89
2.01E-06
136.00

PPSOV
6.43
4.69E-01
12.47
120.6274.32
249.20210.682.51
420.5911.68
038.86

PSOV. PPSO does not have the shared velocity but also havethe plural swarms. Therefore, PSOV have the shared velocity
but also have only the one swarm. The population size M isset to 60 in PSO and PSOV. PPSO and PPSOV have K = 6swarms, and each swarm contains 10 particles, i.e. M = 60.
The inertia weight is fixed as w = 0.6. For PSO and PSOV, theacceleration coeﬃcients are set as c1 = c3 = 1.8 and c2 = 0.
For PPSO and PPSOV, the acceleration coeﬃcients are set as
c1 = 1.8, c2 = 1.6 and c3 = 0.2. For PSOV and PPSOV, theshared velocity coeﬃcients is set as cv = 0.3 is used in the following simulation. We carry out the simulation 30 times for
all the optimization functions with 2000 generations, namelyT = 2000.
The performances with minimum, maximum and meanfunction values on four functions are listed in Table 1. We
can see that the PPSOV can obtain the best mean values for
all the test functions. Furthermore, for f4 , the minimum valueof PPSOV is 0, therefore, only PPSOV was able to find theoptimum function value. Figure 2 shows the mean gbest values of every generation over 30 runs for four test functions.
From these results, we can see that the convergence of PPSO
and PPSOV are quickly. Furthermore, the convergence ofPSOV is not quickly. However, the values decline steadily.
Therefore, we can investigated that the plural swarms prompt
the quick search, and the shared velocity prompt the bust out
from local optima.
From these results, we can say that PPSOV is the eﬀective
algorithm which has the advantage of the plural swarms andshared velocity.

where x ∈ [−600, 600]D and the optimum solution x∗ are all[0, 0, . . . , 0].

4. Conclusions

The optimum function values f (x∗ ) of all functions are 0. f1
and f2 are unimodal functions, and f3 and f4 are multimodal
functions with numerous local minima. All the functions haveD variables, in this study, D is set to 60. We compare the proposed PPSOV to three PSOs; the standard PSO, PPSO andIn this study, we have proposed the new PSO algorithm, PSO containing plural swarms using shared velocity (PPSOV). PPSOV consists of not uni-swarm but pluralswarms. Each particle of PPSOV belongs to one of pluralswarms. The particles belonging to each swarm share the ve-- 205 -341010

Gbest f (Pg)210

Gbest f (Pg)

PSO
PPSO
PSOV
PPSOV

110010

PSO
PPSO
PSOV
PPSOV
31020500

1000
Generation ( t )1500

1020000500(a)1000
Generation ( t )1500

2000(b)3102.8Gbest f (Pg)

10Gbest f (Pg)

PSO
PPSO
PSOV
PPSOV

2.6

10

PSO
PPSO
PSOV
PPSOV
2102.4

10

10500

1000
Generation ( t )1500

102000(c)

0500

1000
Generation ( t )1500

2000(d)Figure 2: Mean gbest value of every generation for 60-dimensional four functions. (a) Sphere function. (b) Rosenbrock’sfunction. (c) Rastrigin’s function. (d) Griewank’s function.

locity (called “shared velocity”) with other particles in theswarm. We have investigated behavior of PPSOV with thesimulation and have confirmed the eﬃciency of PPSOV.

[6] M. Sugimoto, H. Matsushita and Y. Nishio, “ParticleSwarm Optimization Containing Shared Velocity” inIEEE Workshop on Nonlinear Circuit Networks, 2010.

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