2010 IEEE Workshop on
Nonlinear Circuit Networks, Tokushima, Dec 3-4, 2010

Particle Swarm Optimization
Containing Shared Velocity
Masaki SUGIMOTO

Haruna MATSUSHITA

Yoshifumi NISHIO

Dept. Electrical and Electronic Eng.,
Dept. Electrical and Electronics Eng.,
Tokushima University
Hosei University
Email: sugimoto@ee.tokushima-u.ac.jp Email: haruna.matsushita.pd@k.hosei.ac.jp

Abstract— In this study, we propose PSO containing sharedvelocity (PSOV). The important feature of PSOV is that eachparticle of PSOV shares the same velocity information. We
investigate the behavior of PSOV and confirm its eﬃciency.

I. Introduction
Particle Swarm Optimization (PSO) [1] is a popular optimization technique for solving objective functions and PSO
is an evolutionary algorithm to simulate the movement offlocks of birds toward foods. Due to its simple concept, easy
implementation and quick convergence, PSO has attracted
attentions and has been widely applied to diﬀerent fields
in recent years. Furthermore, PSO has demonstrated great
performances for many problems. However, quick convergenceoften leads to local optimum problem. It is important for
multimodal functions with a lot of local optima to compromise
between the quick convergence and being trapped in a local
optimum. In order to escape from such local optima andto avoid the premature convergence, the search for a global
optimum should be diverse.
In this study, we propose an improved PSO algorithm; PSO
containing cooperative particles (called PSOV).
We explain the algorithm of PSOV in detail in Section II. In
Section III, we perform basic numerical experiments by using
two algorithm methods of PSO and PSOV. Furthermore, weconfirm the eﬃciency of PSOV.

Dept. Electrical and Electronic Eng.,
Tokushima University
Email: nishio@ee.tokushima-u.ac.jp

where r1 and r2 are two random variables distributed uniformly
on [0, 1], w is an inertia weight of all particles, and c1 and c2
are positive acceleration coeﬃcients.
[PSOV4] All the particles shared V c = (vc1 , vc2 , ..., vcD ). Let
V c represent the average velocity of all the particles;
M1 ∑(2)
vcd =vid
M i=1[PSOV5] Update Xi depending on its V c and V i ;xid (t + 1) = xid (t) + cv vcd (t + 1) + vid (t + 1),(3)where cv is a cooperation coeﬃcients. In other words, theparticles combine the action of individual and cooperation.
[PSOV6] Let t = t + 1. Go back to [PSOV2], and repeat untilt = T.
III. Numerical Experiments
In order to confirm the performance of PSOV algorithm, we
have performed basic numerical experiments. The problem isfinding the optimum (minimum) value of f (x) in the algorithm.
Referring to [2], we use the following four bench marks.
1. Sphere function:
D−1∑
f1 (x) =xd2 ,(4)d=1where x ∈ [−2.048, 2.047] and the optimum solution x∗ are
all [0, 0, . . . , 0].
DII. PSOV[PSOV1] (Initialization) Let a generation step t = 0. Randomly initialize the particle i = (1, 2, ..., M) position Xi =(xi1 , xi2 , ..., xiD ) and its velocity V i = (vi1 , vi2 , ..., viD ) for allparticles i and initialize Pi = (pi1 , pi2 , ..., piD ) with a copy ofXi .
[PSOV2] Evaluate the current cost f (Xi ). Update the personal
best position pbest Pi = (pi1 , pi2 , ..., piD ) for each particle i andthe global best position gbest Pg = (pg1 , pg2 , ..., pgD ) among theall particles.
[PSOV3] Update V i of each particle i depending on its pbest
and its swarm best gbest;vid (t + 1) = wvid

+c1 r1 {{pid − xid (t)}}+c2 r2 pgd − xid (t) ,(1)2. Rosenbrock’s function:
D−1∑f2 (x) =(100(xd2 − xd+1 )2 + (1 − xd )2 ),

(5)

d=1

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):D∑f3 (x) =(xd2 − 5 cos(2πxd ) + 5),

(6)

d=1

where x ∈ [−5.12, 5.12]D and the optimum solution x∗ are
all [0, 0, . . . , 0]. We consider that an almost optimum value- 74 -

10

41010PSO
PSOV

010PSOPSOV

gGbest f (P )Gbest f (Pg)3−1010−201010

210−3010−4010105001000Generation ( t )1500

1020000500(a)1000
Generation ( t )1500

2000(b)331010PSOPSOVPSO
PSOVgGbest f (P )Gbest f (Pg)

221011010

11000500

1000
Generation ( t )1500

1020000500(c)1000Generation ( t )

1500

2000

(d)

Fig. 2. 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.

TABLE I
10Comparison results PSO and PSOV on test functions with D = 30.

10cv = 0.20gGbest f (P )fcv = 0.3

10cv = 0.4
−10f1

−20

f2−30

f3

101010−40

10Fig. 1.

0500

1000
Generation ( t )

1500

f4

2000

PSO
PSOV
PSO
PSOV
PSO
PSOV
PSO
PSOV

Avg.
1.67E-04
5.92E-34
43.8
24.3
80.6
38.4
1.21E+01
3.04

Min.
3.33E-35
3.18E-39
20.2
21.2
46.8
19.9
3.00E-15
0Max.
4.73E-03
1.69E-32
132.1
28.4
117.4
130.7
9.09E+01
9.07E+01

The cv of various value is compared. (Sphere function, D = 30)

is obtained if the algorithm attains the criterion f1 (x) = 100.
This criterion is based on [2]. The range of initialization is−5.12 ≤ xi ≤ 5.12.
4. Griewank’s function:DD∑∏xd2xdcos( √ ) + 1,f4 (x) =+4000 d=1dd=1(7)where x ∈ [−600, 600]D and the optimum solution x∗ are all[0, 0, . . . , 0].
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 have

D variables. In this study, D is set to 30 and 100 to investigate
the performances in various dimensions. The population size
M is set to same the D. The inertia weight is fixed as w = 0.5.
For PSO and PSOV, the acceleration coeﬃcients are set as
c1 = c2 = 1.8. We experimented in how many influences it
has on a simulation by changing the value of cv We compare
PSOV in the cv , Sphere function was used by D = 30 this time.
Because the best result was brought from Figure. 1 at the timeof cv = 0.3, cv = 0.3 is used in the following simulation. For
PSOV, the cooperation coeﬃcients are set as cv = 0.3.
We carry out the simulation 30 times for all the optimization
functions with 2000 generations, namely T = 2000. The
performances with minimum, maximum and mean function
values on four functions with 30-dimension are listed inTable I. We can see that the PSOV can obtain the best

- 75 -

551010
PSO
PSOV

PSO
PSOV
410g

Gbest f (P )Gbest f (Pg)010−510310210−1010

10500

1000
Generation ( t )1500

1020000500(a)1500

2000(b)441010PSOPSOVPSO
PSOV

310g

Gbest f (P )Gbest f (Pg)

1000
Generation ( t )310210110010210−10500

1000
Generation ( t )1500

102000

(c)

0500

1000
Generation ( t )

1500

2000

(d)

Fig. 3. 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.

TABLE II

Because it is diﬃcult for PSO to find the optimum solution,
PSO are easily trapped in the local optima and prematurely
converge.
From these results, we can confirm that the mean values ofPSOV are the best among four problems. Therefore, we canconfirm that PSOV algorithm is more eﬀective than standard
PSO.

Comparison Results PSO and PSOV on test functions with D = 100.

ff1
f2
f3
f4

PSO
PSOV
PSO
PSOV
PSO
PSOV
PSO
PSOV

Avg.
4.12E-02
3.52E-08
165.8597.60
432.84186.855.09E-01
1.07E-01

Min.
4.68E-09
6.13E-10
88.67
92.64
358.18104.47
2.28E-06
2.28E-07

Max.
1.18
3.90E-07
251.85151.41603.94270.63
4.13
1.15

IV. Conclusions

mean values for all the test functions. Next, in case of theperformances on the test functions with D = 100 shown inTable II, PSOV can obtain the best mean values for all thetest functions.
Figs. 2 shows the mean gbest values of every generation
over 30 runs for four test functions with 30 dimensional.
From these results, we can see, PSOV can obtain the bestresults. However, the performances of PSO for the Rastrigin’sfunctions f3 as Figs. 2(c) is more good value until 500generation. From this results, we have investigated that sharedvelocity is helping to escape from local minima by the block
the quick convergence.
Meanwhile, the mean gbest values for 100 dimensional test
functions are shown in Fig. 3. In whole generation steps, theperformances of PSOV is the good values in all the functions.

In this study, we have proposed a new Particle Swarm Optimization algorithm (PSOV) containing the containing sharedvelocity. We have performed the basic numerical experimentsfor various dimensions by using the PSO and PSOV. Fromthese results, we have confirmed that PSOV could obtain theeﬀective.
References[1] J. Kennedy and R. Eberhart, “Particle swarm optimization,” Proc. IEEE Int. Conf. on Neural Networks, vol. 4,pp. 1942–1948, 1995.
[2] J. Kennedy and R. Medes, “Population structure andparticle swarm performance,” Proc. IEEE Congress on
Evolutionary Computation, vol. 2, pp. 1671–1676, 2002.

- 76 -