2009 International Workshop on Nonlinear Circuits and Signal Processing
NCSP'09, Waikiki, Hawaii, March 1-3, 2009

Particle Swarm Optimization Containing Plural Swarms
Masaki Sugimoto, Taku Haraguchi, Haruna Matsushita and Yoshifumi Nishio
Tokushima University, Japan
Phone:+81-88-656-7470, Fax:+81-88-656-7471{sugimoto, taku, haruna, nishio}@ee.tokushima-u.ac.jp

Abstract

The feature of PPSO is that its swarm is not one but plural.
The swarms share an information of the best position in eachswarm. Except the swarm including the best particle in wholeswarms, all the particles are repositioned to escape the local
optima. We confirm the eﬀectiveness of the PPSO algorithm
by numerical experiments.

In this study, we proposes a modified particle swarm optimization (PSO) called plural PSO (PPSO) for bust-out the local optimum solution. The feature of PPSO is that the swarm
of PPSO is not one but plural. The plural swarms share an
information of the best position in each swarm. Except theswarm including the best particle in whole swarms, all the
2. Plural PSO (PPSO) Algorithm
particles are repositioned to escape the local optima. We investigate behaviors of PPSO and confirm its eﬃciency in mulThe most important feature of PPSO is that PPSO has K
timodal functions.
swarms, i.e. plural particle groups. Each swarm is denoted as
S k (k = 1, 2, . . . , K) and consists of N/K particles. The position vector of each particle i and its velocity vector are rep1. Introduction
resented by Xi = (xi1 , xi2 , . . . , xiD ) and V i = (vi1 , vi2 , . . . , viD ),Particle swarm optimization (PSO) [1] is a popular opti- respectively, where (i = 1, 2, . . . , N) and xid ∈ [xmin , xmax ].
mization technique for the solution of object function and is [PPSO1](Initialization) Let t = 0 and t = 0, i.e., t is theRan evolutionary algorithm to simulate the movement of flocks simulation step and t is the time step for the reposition. RanR
birds and the movement of a school fishes toward foods. Due domly initialize the particle position X and its velocity V foriito the simple concept, such as easy implementation and quick all particles i and initialize P = (p , p , . . . , p ) with a copy
i
i1 i2
iD
convergence, PSO has gained much attentions and wide ap- of X . Evaluate the objective function f (X ) for all particle iiiplications in diﬀerent fields in resent years. However, PSO and find P with the best function value among the all partig
has demonstrated a great performance for many problems cles. Attach each particle i to any swarm S at random andkyet its fast convergence often leads to premature convergence find P = (p , p , . . . , p ) with the best function value
sk
sk 1
sk 2
sk D
in local optima. The tradeoﬀ between fast convergence and among the particles belonging to the swarm S ;kbeing trapped in local optima will be even more critical insk = arg min{ f (Xi )},i ∈ S k.
(1)
multimodal functions having many local optima very closeito each other. In order to escape from the local optima andavoid premature convergence, the search for global optimum
should be diverse. Many researchers have improved the per- [PPSO2] Evaluate the current cost f (Xi ). Update the performance of PSO by enhancing its ability with a more diverse sonal best position (called pbest) Pi for each particle i andsearch [2]-[6]. Specifically, some researchers have proposed the global best position gbest Pg among the all particles in
the PSO using multiple swarms, and exchange information all the swarms so far (called gbest). For each particle i belonging to each swarm S k , update the swarm best position
among them [7], [8].
Meanwhile, in the real world, a large company reshuﬄes P sk with the best cost among the particles belonging to thethe personnel regularly. Furthermore, the company members swarm S k so far (called sbest), if needed.
work not as a group but as a organization consisting of some [PPSO3](Update) Update the velocity vector V i and the podepartments. Therefore, they can work more eﬀective by ex- sition Xi of each particle i depending on its pbest, gbest andchanging informations among the departments in the com- sbest according topany.
vid (t + 1) = wvid (t) + c1 r1 {pid {− xid (t)} }In this study, we propose a new PSO algorithm contain{}+ c2 r2 p sk d − xid (t) + c3 r3 pgd − xid (t) , i ∈ S k , (2)ing plural swarms (called PPSO). The proposed algorithm reflects the strategy of the firm for the business in the real world.
xid (t + 1) = xid (t) + vid (t + 1),- 419 -

where d = 1, 2, . . . , D, r1 , r2 and r3 are three random variables
distributed uniformly in [0, 1], w is an inertia weight, and c1 ,
c2 , and c3 are positive acceleration coeﬃcients.
[PPSO4] If tR = T/K, we perform [PPSO5], if not, we perform [PPSO6]. Thus, we perform [PPSO5] every time when
T/K simulation steps are performed. T is the maximum number of the simulation.
[PPSO5](Reposition) Reposition the positions Xi of the particles belonging to all the S k except the best swarm at random
and reassign their velocities V i at random. We reset tR = 0.
[PPSO6] Let t = t + 1 and tR = tR + 1. Go back to [PPSO2],
and repeat until t = T .
3. Numerical Experiments
In order to confirm the performance of PPSO algorithm, we
carry out performed basic numerical experiments. The goal
is to find the optimum (minimum) value of f (x). Referring
to [2], we use the following four benchmark functions.
1. Sphere function:
f1 (x) =

D−1
∑

xd2 ,

(3)

d=1

where 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 ),

(4)

d=1

Table 1: Comparison results PSO1, PSO2 and PPSO on test
functions with D = 30.
f Methods
Mean
Minimum Maximum
PSO1
7.92e-40 2.53e-43
7.61e-39
f1
PSO2
6.62e-29 4.26e-31
4.74e-28
PPSO
2.84e-54 4.33e-56
2.84e-53
PSO1
25.08
3.64
76.45
f2
PSO2
23.92
0.34
90.84
PPSO
24.94
3.37
80.68
PSO1
27.92
10.89
40.59
f3
PSO2
16.04
6.93
32.67
PPSO
12.67
6.93
19.80
PSO1
1.15e-02
0
4.18e-02
f4
PSO2
8.20e-0305.15e-02
PPSO
3.29e-04
09.86e-03

Table 2: Comparison results PSO1, PSO2 and PPSO on test
functions with D = 100.
f Methods
Mean
Minimum Maximum
PSO1
8.01e-05
1.66e-06
1.21e-03
f1
PSO2
1.33e-05
1.72e-06
4.16e-05
PPSO
3.57e-04
1.13e-05
1.29e-03
PSO1
6.17e+04 1.54e+02 9.24e+05
f2
PSO2
3.42e+04 2.06e+02 8.42e+05
PPSO
2.87e+02 1.61e+02 4.34e+02
PSO1
258.63208.88324.70
f3
PSO2
188.42121.97247.79PPSO
123.6291.08
167.32
PSO1
3.53e-03
2.05e-08
2.22e-02
f4
PSO2
3.94e-03
4.08e-08
2.46e-02
PPSO
2.41e-03
6.51e-07
4.64e-02

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

(5)

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) =

D
D
∑
∏
xd2
xd
cos( √ ) + 1,
+
4000
d
d=1
d=1

(6)

where x ∈ [−600, 600]D and the optimum solution x∗ are all[0, 0, . . . , 0].
The optimum function values f (x∗ ) of all function are 0. f1

and f2 are unimodal functions, and f3 and f4 are multimodal
functions with numerous local minima. All the functions have
D variables. In this study, D is set to 30 and 100 to investigate
the performances in various dimensions.
In order to evaluate the eﬃciency of PPSO and to investigate behaviors of PPSO, we compare the three algorithms that
PSO1, PSO2 and PPSO. PSO1 is the standard PSO and PPSO
is the proposed algorithm explained in Section 2. PSO2 is thestandard PSO with only the reposition process of PPSO.
The population size N is set to 40 in PSO1 and PSO2.
PPSO have K = 4 swarms, and each swarm contains 10 particles, i.e. N = 40. For PSO1, PSO2 and PPSO, the inertia
weight is fixed as w = 0.6. For PSO1 and PSO2, the acceleration coeﬃcients are set as c1 = c2 = 1.8. For PPSO, theparameters are set as c1 = 1.8, c2 = 1.4 and c3 = 0.4. The
timing of reposition is set as tR = 500.
We carry out the simulation 30 times for all the optimization functions with 2000 generations, namely T = 2000. The- 420 -

6

10

10

10

PSO1
PSO2
PPSO

Function Value log(f(x))

−10

10

PSO1
PSO2
PPSO

5

10
Function Value log(f(x))

0

10

−20

10

−30

10

−40

10

4

10

3

10

2

10
−50

10

1

−60

10

0

500

1000

1500

10

2000

0

500

(a)
PSO1
PSO2
PPSO

10
Function Value log(f(x))

Function Value log(f(x))

10

2

10

10

10

10

1

0

500

1000

1500

2000

(b)

3

10

10

1000
Number of Generation t

Number of Generation t

1500

2000

10

Number of Generation t

1

PSO1
PSO2
PPSO

0

−1

−2

−3

−4

0

500

1000

1500

2000

Number of Generation t

(c)

(d)

Figure 1: Mean gbest value of every generation for 30-dimensional four functions. (a) Sphere function. (b) Rosenbrock’s
function. (c) Rastrigin’s function. (d) Griewank’s function.

performances with minimum, maximum and mean function
values on four functions over 30 runs with 30-dimension arelisted in Table 1. We can see that the mean values of PPSO
are the best in f1 , f3 and f4 . However, in case of the performances on the test functions with D = 100 shown in Table 2,
PPSO are the best in f2 , f3 and f4 . From these results, we cansay that PPSO is more eﬀective for multimodal functions.

tion sbest. Therefore, PPSO can find better position quickly
in early stage of the optimization. On the other hand, we
can see that the graphs of PSO2 and PPSO are waving by theeﬀect of the reposition to search better position. However,
because the performances of the proposed PPSO are better
than PSO2, we can say that PPSO, which exchanges the information between some swarms, is more eﬀective than PSO2
which is the standard PSO with only the reposition process.
Figure 1 shows the mean gbest value of every generation In other words, these eﬀects avert the premature convergenceover 30 runs for four test functions with 30 variables. The and the particles of PPSO easily escape from the local optiperformances of PSO1 for the all functions do not achieve mum.
the good solution, instead, the proposed PPSO can obtain the
From these results, we can confirm that PPSO is more efbest results 3 times. Meanwhile, the mean gbest values for
fective for multimodal function.
100 dimensional test functions are shown in Fig. 2. The performances of PPSO for the function f1 as Figs. 1(a) is theworst values in the three algorithms. However, for the func- 4. Conclusions
tions f2 , f3 and f4 as Figs. 2(b)–(d), PPSO can obtain thebest results. In particular, PPSO greatly improves the perIn this study, we have proposed the new PSO algorithm,
formance from PSO1 on f1 , f3 and f4 with D = 30, and f2 Plural PSO (PPSO). PPSO has plural swarms, and all theand f3 with D = 100. Because PPSO has plural swarms and swarms share the information of their the best position. We
searches with sharing an information of the swarm best posi- have investigated its behaviors with the simulation and have

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Function Value log(f(x))

10

10

10

10

10

7

4

10

PSO1
PSO2
PPSO

2

10

0

−2

5

10

4

10

3

−4

10

2

−6

0

PSO1
PSO2
PPSO

6

Function Value log(f(x))

10

500

1000

1500

10

2000

0

500

(a)
PSO1
PSO2
PPSO

10
Function Value log(f(x))

Function Value log(f(x))

10

3

10

10

10

2

0

500

1000

1500

2000

(b)

4

10

10

1000
Number of Generation t

Number of Generation t

1500

2000

10

Number of Generation t

1

PSO1
PSO2
PPSO

0

−1

−2

−3

0

500

1000

1500

2000

Number of Generation t

(c)

(d)

Figure 2: Mean gbest value of every generation for 100-dimensional four functions. (a) Sphere function. (b) Rosenbrock’s
function. (c) Rastrigin’s function. (d) Griewank’s function.

confirmed the eﬃciency of PPSO.

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