�

Abstract—In this paper, an advanced PSO has been proposed to solve multi-modal functions optimization. Multiple swarms are used to optimize parallel search and artificial repulsive potential field on local points are set up to prevent multiple swarm converging to the same place and repeatedly search. In addition, this paper provides a theoretical basis for the strategyof multi-swarm parallel search used in the advanced algorithm. Finally, the method is tested on some benchmark functions and the results show a superior performance compared to the others PSO variant.

I. INTRODUCTION

article Swarm Optimization is a stochastic optimization based on swarm intelligence. For its fast convergence speed and less parameters, PSO algorithm has been

concerned by many scholars recently [1]. It is often used to solve nonlinear, unsmooth and multi-modal function optimization, and now it has been widely used in many scientific and engineering fields.

However, in practice, people find that fast loss of

population diversity during the evolution easily causes PSO

sunk into local extremum and leads to PSO early convergence

when solving some multi-modal smooth functions. To solve

the above mentioned problem, many scholars propose

improving methods, for example, the increase of particles

diversity through introduction of evolutionary selection

mechanism [2] or spatial neighborhood dynamic adjustment

[3].

Multi-swarm cooperation can increase particle diversity

and accelerate convergence speed of PSO.Shi and Krohling

proposed a co-evolutionary algorithm based on two PSO to

solve the problem of max-min values [4]. Niu et al. Suggest

the new multi-swarm cooperative to solve the function

optimization [5]. Also a cooperative model with two layers

Manuscript received June 23, 2011. The Project was supported by the Special Fund for Basic Scientific Research of Central Colleges, China University of Geosciences (Wuhan). CUGW090206.

Qin Tang is with the Department of Control Science and Engineering,Huazhong University of Science and Technology, and work in School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, P. R. China (e-mail: tq7171@gmail.com).

Yi Shen is with the Department of Control Science and Engineering,Huazhong University of Science and Technology, Wuhan 430074, P. R.China (e-mail: yishen64@163.com).

Chengyu Hu is with the School of Computer, China University of Geosciences, Wuhan 430074, P. R. China (phone: 86-15927160586; e-mail: huchengyu@cug.edu.cn).

Jianyou Zeng is with the School of Arts and Communication, China University of Geosciences, Wuhan 430074, P. R. China (e-mail: jianyou@cug.edu.cn).

framework was proposed by Li [6], Wang et al, Proposed

multi-swarm co-evolutionary algorithm applied to Neural

network noise canceller and the training of RBF neural

network’s structure and parameters [7] [8].

In addition, multi-swarm cooperative co-evolutionary is

widely used to solve dynamic optimization problems.

Blackwell proposed Multi-swarm PSO which is to keep the

population with more diversity through the anti-convergence

to prevent the population meeting together [9].

Speciation-based PSO, advanced by Parrot, is used for

optimization of multi-modal functions and dynamic

optimization problem [10]. Recently, the application of PSO

to dynamic problems through multiple swarms

co-evolutionary has also been explored [11][12]. The paper

[13] shows that the cooperative technique can also solve large

scale problem.

Though multi-swarms can increase the population diversity

and accelerate convergence speed, we cannot jump to the

conclusion that multi-swarm parallel search is more efficient

than single-swarm search. Some mathematical quantitative

proof is still needed to be given. Additionally, in practice,

multi-swarm likely does repeated search to a partial area and

cause algorithm inefficiency. Therefore, in this paper a

multi-swarm cooperation PSO in repulsive potential field has

been put forward and the main idea of this algorithm is to set

up artificial potential field at local extreme points to prevent

multiple swarms repeated search and to explore the new area.

At the same time, in order to assure the convergence precision,

one swarm is left to exploit discovered local extreme point.

II. MULTI-SWARMS COOPERATION OPTIMIZATION

In biosphere, not only exists Darwin’s natural evolution law

—“survival of the fittest”, but also communal evolution law

which insists that multiple individuals or species co-evolve

through inter-cooperation. Cooperation evolution algorithm is

just originated from this thought.

Multi-swarm cooperation can be classified into

Competitive Co-evolution and Cooperative Co-evolution and

the latter one is adopted in this paper. Cooperative

Co-evolution is generally defined as: multiple swarms (or

sub-swarms) searching for a solution (serially or in parallel)

and exchanging some information during the search according

to some communication strategy. Based on the exchanged

information, an action is taken to effectively continue with the

search process. Cooperative Co-evolution is a kind of

macro-coevolution method.

Multi-swarm Cooperation Optimization for Multi-modal Functionsin Repulsive Potential Field

Qin Tang, Yi Shen, Chengyu Hu and Jianyou Zeng

P

70

Fourth International Workshop on Advanced Computational Intelligence Wuhan, Hubei, China; October 19-21, 2011

978-1-61284-375-9/11/$26.00 @2011 IEEE

The most important step of multi-swarm cooperation

optimization is to decompose the problem or space. From the

point of space division, cooperative co-evolution includes

implicit and explicit space division. Explicit space

decomposition divides the space directly according to

correlation of dimension, for example, dividing n dimension

space into n spaces and then using n swarms to optimize

respectively and finally putting the optimum individual

together to form a solution vector. Implicit space division

means that multi-swarms search the whole space at the same

time. Actually, each swarm search different areas because of

different initialization and parameters.

Several papers have proved that multi-swarm cooperation

is better than single swarm search through simulation [14], but

there is no mathematics analysis till now. This paper presents

some assumptions firstly and proves that multi-swarm

cooperation is better than single swarm search under these

assumptions.

Assumption: if searching space is a d dimension hyper

sphere with radius R, there are K local extreme point areas

with radius r1,…,rk marked as m1,m2,…,mk, One of the local

extreme point is x* of the global extreme points and the area is

m*. Multi-swarm cooperation search adopts implicit space

division.

Theorem: if the assumption is tenable, multiple swarms

have a higher probability finding a global optimal solution by

co-evolution than single swarm does.

Proof: to single swarm search, { *}sP x m� means optimal

solution probability of single swarm search result and the probability is

1

{ *} { }K

s s k

k

P x m p x m�

� � �� (1)

The probability of mk varies directly as the radius of this area and the probability is

{ }

d

s k krp x mR

� � � �� �

(2)

According to (1) and (2), the probability of extreme points in single swarm search result is

� 1 2

1

{ *}

ddKs Kk

dKk

r r rrP x m

R R�

� � �� � �� �

� �

� (3)

To multi-swarm cooperation search, if there are K swarms, implicit space division is used and each swarm search corresponding local extreme point area. The maximum range of each swarm search is Rk, and Rk<R, that is

1 2

K

kR R R R� � � � (4)

The probability of extreme points in multi swarm search

result is { *}mP x m�

1 2

1 11 2

{ *} { }

d dK K

m s k k K

k k k K

r r r rP x m p x m

R R R R� �

� � �� � � � �

� � �

� � � �� �

� �� � (5)

So

� 1 2

m K d

ds

K

P R

P R R R

�

�� � �

(6)

As (1) and 1d � we know 1m

s

P

P� , namely

( * *) ( * *)m sP x m P x m� � � (7)

To sum up, if the assumption is tenable, the probability of optimal value searched by multi-swarm is higher than the single swarm does.

III. MULTI-SWARM COOPERATION OPTIMIZATION IN REPULSIVE

POTENTIAL FIELD

In this paper, an advanced cooperative PSO was proposed. More than two swarms referred to as groups are used, and all performing the same PSO algorithm. The exchange of information was also performed every number of iterations. And the gbest of every group is shared. As has shown before, we decompose the search space in an implicit way, every group search in parallel. Unlike the other cooperative PSO, we set up some repulsive point in the search area.

As is known that when use multi-swarm to optimize the

multimodal function in parallel, multiple swarms probably

plunge into a local extreme point. To solve this problem, one

way is that order the fitness value of the swarms which sink

into a local extreme point and keep the best swarm continuing

to search and the others are initialized randomly. The

drawbacks of this method are that initialized swarm or swarms

which did not sink into local extreme point have a high

probability to search repeatedly in the area of local extreme

point. As a result, computing efficiency reduces greatly.

In this paper, setting artificial repulsive potential field at local extreme point can prevent repeated search to local extreme points discovered by multi-swarms. The first step of this algorithm is to group swarms randomly and search in parallel. If three or more swarms cluster, we know this area exist local extreme points, then we set repulsive potential field in the point. In repulsive potential field, swarm with the best fitness value residents and does precision exploitation search; other swarms initialize randomly and explore in searching space. The flowchart of new algorithm is presented as figure 1.

The formula of setting up artificial repulsive potential field in local extreme point as (8).

2

1 1 1

( ) 2

0

ore o

U� � �

� � �

� �

� ��

�

� � �� � � ���

(8)

� is position gain coefficient; � is distance from the swarm

out of repulsive potential field to local extreme point; �0 is

repulsive radius which is a fixed value defined before in the

hyper sphere. Here the distance refers to the Euclidean

distance.

71

Fig. 1. The flowchart of RC_PSO.

IV. EXPERIMENTS

A. Experimental Frameworks

In order to verify the validity of improved algorithm, the

publicly benchmark function Griewank, Ackley and Rastrigin

are tested respectively. These three benchmark functions are

complicated, nonlinear, multimodal functions and own a large

number of extreme points. They can reflect the abilities of the

new algorithm in keeping swarm diversity, searching and

escaping from the local extreme points.

The three benchmark functions are:

Ackley function:

2 1

1 1

11 20 20exp( 0.2 ) exp( cos(2 ))

N N

n nNn n

F e x xN

�� �

� � � � �� � (9)

Griewank function:

� 2

11

12 1 cos

4000

NN

n n nn

F x x n��

� � � �� (10)

Rastigrin function:

2

1

3 [ 10cos(2 ) 10]N

n nn

F x x��

� � �� (11)

The global optimal point of these three benchmark

functions are X*= 0,0, 0 , F(X*)=0 Parameter setting

of benchmark function presents in table 1.

TABLE 1 BENCHMARK FUNCTION PARAMETERS

Benchmark function

Range of Searching space

Max gen size dim

Ackley [-32.768,32.768] 3000 10 10

Griewank [-600,600] 3000 10 10

Rastrigin [-5.12,5.12] 3000 10 10

To know better about the performance of improved

algorithm, this paper compares RC_PSO with PSO_w,

FDR_PSO, FIPS, UPSO and CLPSO. Each algorithm runs 20

times and algorithm parameters keep the same with references

[15]. Inertial weights are 0.9~0.4, acceleration factors are

C1=C2=2. The swarm size is 10. the dimension of search

place is 10. the maxim evolution generation is 3000 and

precision of solution is 10-40. The fixed value of arrogation

radius in RC_PSO is 0.01. This means that when the distance

among three or more swarms is below 0.01, a local extreme

point must exist. The radius of repulsive potential field is 0.1.

B. Simulation Results

The simulation results are presented in Table II.

TABLE II SIMULATION RESULTS BY FIVE VARIANTS OF PSO

FDR_PSO FIPSO UPSO CLPSO RC_PSO

F1best 7.105e-15 3.552e-15 3.552e-15 7.105e-15 3.552e-15mean 0 0 0.7796 0 0 worst 7.105e-15 2.21e-14 2.0133 3.552e-15 3.552e-15

F2best 0.0282 0.00939 0.022141 1.0396e-6 0 mean 0.1208 0.0568 0.0876 0.0090 0.006worst 0.2653 0.13933 0.19231 0.014806 0.13389

F3best 0 0.99496 3.9798 0 0 mean 6.3677 3.1213 14.8683 0.0995 0 worst 12.934 6.9647 31.839 0.99496 1.760e-11

From Table II we can see that mean values, the best

values and the worst values of RC_PSO are better than the

other variants of PSO. To Ackley function, UPSO is easy to

sink into local extreme points. In the 20 times’ runs, all the

variants can find the global extreme points except UPSO. To

Griewank function, CLPSO and RC_PSO work better.

RC_PSO always find the best value. To Rastrigin function, the

four classic variants are difficult to find the optimal value, but

the proposed RC_PSO performs best which indicates the

improved algorithm owns the ability of jumping out of local

extreme points and fast convergence speed.

Figure 2~4 are the curve of fitness value of five algorithms test on three benchmark functions.

Updating speed and position of each group, computing fitness value

Swarm initialization

Updating pbest and gbest according to fitness

Producing Local extreme points, settingrepulsive potential field

Dividing and grouping

Swarms resident at local extreme points do local exploitation search and other swarms do large range exploration search by weights dynamic

change in repulsive potential field

Satisfied?

N

Y

Swarm aggregation and convergence

Y

N

Stop

72

0 500 1000 1500 2000 2500 30001E-161E-151E-141E-131E-121E-111E-101E-91E-81E-71E-61E-51E-41E-30.010.1

110

1001000

best

fitn

ess

valu

e

gen

FDR_PSO FIPS UPSO CLPSO RC_PSO

UPSO

FIPS

FDR_PSO

RC_PSO

CLPSO

Fig. 2. Fitness value curves tested on Ackley function

0 500 1000 1500 2000 2500 3000

0.01

0.1

1

10

100

best

fitn

ess

valu

e

gen

FDR_PSO FIPS UPSO CLPSO RC_PSO

CLPSOUPSO

FDR_PSOFIPS

RC_PSO

Fig. 3. Fitness value curves tested on Griewank function

0 500 1000 1500 2000 2500 30001E-171E-161E-151E-141E-131E-121E-111E-101E-91E-81E-71E-61E-51E-41E-30.010.1

110

1001000

10000

best

fitn

ess

valu

e

gen

FDR_PSO FIPS UPSO CLPSO RC_PSO

RC_PSO

UPSO

CLPSO

FIPS

FDR_PSO

Fig. 4. Fitness value curves tested on Rastrigin function

From the fitness value curves in Fig.2~4 we can conclude

that convergence speed of RC_PSO is in the middle of all

algorithms; to Ackley and Griewank function, the

convergence speed of FIPS is the fastest. The proposed

algorithm scarifies certain searching speed for getting a higher

convergence precision. In general, the search ability of

RC_PSO is better than the other algorithms but its

convergence speed is in the middle.

When the dimension of the searching space increased,

RC_PSO still performs better than other algorithm. In the

following figure 5-7, the dimension of the test function is 100,

Parameter setting of benchmark function presents in table 3.

TABLE 3BENCHMARK FUNCTION PARAMETERS

Benchmark function

Range of Searching space

Max_gen size dim

Ackley [-32.768,32.768] 1000 30 100

Griewank [-600,600] 1000 30 100

Rastrigin [-5.12,5.12] 1000 30 100

0 5000 10000 15000 20000 25000 30000

2

4

6

8

10

12

14

16

18

20

22 FDR_PSO FIPS UPSO RC_PSO CLPSO

fitne

ss v

alue

FEs

Ackly dim=100

RC_PSO

Fig.5. Fitness value curves tested on Ackley function as dim=100

0 5000 10000 15000 20000 25000 30000

0

500

1000

1500

2000

2500

3000

FDR_PSO FIPS UPSO RC_PSO CLPSO

best

fitn

ess

valu

e

FEs

Griewank dim=100

RC_PSO

Fig.6. Fitness value curves tested on Griewank function as dim=100

0 5000 10000 15000 20000 25000 30000-200

0

200

400

600

800

1000

1200

1400

1600

1800 FDR_PSO FIPS UPSO RC_PSO CLPSO

best

fitn

ess

valu

e

FEs

Rastrigin dim=100

RC_PSO

Fig. 7. Fitness value curves tested on Rastrigin function as dim=100

From the figures 5-7, we can draw the conclusion that when

the dimension of the search space increase ,all the algorithms’

73

search ability deteriorate, but comparing with some other

classical PSO variants, RC_PSO can results in better optima,

is more roust and prevents more effectively the premature

convergence.

V. CONCLUSIONS AND THE FUTURE WORKS

Based on the analysis of multi-swarm cooperation

optimization for multi-modal functions, this paper tries to

overcome its shortcomings with the proposition of

multi-swarms cooperation optimization in repulsive potential

field. Though the dimension of the search space increase,

tested on three multi-modal benchmark functions, our

proposed RC_PSO is proved better than the other four

classical algorithms.

Over all, we can conclude that our approach is suitable to multi-modal functions, being robust and outperform than the other variants. However, there is some deficiency in setting aggregation radius and repulsive radius in potential field, this could perhaps be alleviated by making the radius self-adaptive, and this is future work.

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