NESTING DISCRETE PARTICLE SWARM OPTIMIZERS FOR

MULTI-SOLUTION PROBLEMS

Masafumi Kubota and Toshimichi Saito

Faculty of Science and Engineering, Hosei University, 184-8584 Tokyo, Japan

Keywords:

Swarm inteligence, Discrete particle swarm optimizers, Multi-solution problems.

Abstract:

This paper studies a discrete particle swarm optimizer for multi-solution problems. The algorithm consists of

two stages. The ﬁrst stage is global search: the whole search space is discretized into the local sub-regions

each of which has one approximate solution. The sub-region consists of subsets of lattice points in relatively

rough resolution. The second stage is local search. Each subregion is re-discretized into ﬁner lattice points

and the algorithm operates in all the subregions in parallel to ﬁnd all approximate solutions. Performing basic

numerical experiment, the algorithm efﬁciency is investigated.

1 INTRODUCTION

The particle swarm optimizer (PSO) is a simple

population-based optimization algorithm and has

been applied to various systems (Wachowiak et al.,

2004) (Garro et al., 2009) (Valle et al., 2008): signal

processors, artiﬁcial neural networks, power systems,

etc. The PSO has been improved and evolved in or-

der to expand the search function for various prob-

lems (Engelbrecht, 2005): multi-objective problems,

multi-solution problems (MSP), discrete PSO, hard-

ware implementation, etc. The MSP is inevitable in

real systems and several methods have been studied

(Parsopoulos and Vrahatis, 2004). The discrete PSO

operates in discrete-valuedsearch space (Engelbrecht,

2005), (Sevkli and Sevilgen, 2010)and has several ad-

vantages on reliable operation, reproducible results,

robust hardware implementation, etc. However, there

exist not many works of digital PSO for the MSP.

This paper presents the nesting discrete parti-

cle swarm optimizers (NDPSO) for the MSP. The

NDPSO consists of two stages. The ﬁrst stage is

the global search. The whole search space is dis-

cretized into the local subregions (LSRs) that consist

of subsets of lattice points in relatively rough resolu-

tion. Applying the particles with ring topology, the

NDPSO tries to ﬁnd the LSRs each of which includes

the ﬁrst approximate solution (AS1) that is a solu-

tion candidate. The second stage is the local search

where each LSR is discretized into ﬁner lattice points.

The NDPSO operates in all the LSRs in parallel and

tries to ﬁnd all the approximate solutions (AS2). The

roughness in the global search is a key to ﬁnd all AS2s

and the parallel processing in the local search is basic

for effective computation. Since the NDPSO adopts

subdivision, we have used the word ”nesting”. The

NDPSO may be regarded as a discrete version of the

multi-population method (Yang and Li, 2010) with an

eagle strategy (Yang and Deb, 2010). Performing ba-

sic numerical experiments, the algorithm capability is

investigated.

2 NESTING DISCRETE PSO

For simplicity, we consider the MSP in 2-dimensional

objectivefunctionsF(x) ≥ 0, x ≡ (x

1

,x

2

) ∈ R

2

where

the minimum (optimal) value is normalized as 0. F

has multiple solutions x

i

s

, i = 1 ∼ M: F(x

i

s

) = 0,

x

i

s

= (x

i

s1

,x

i

s2

) ∈ S

0

. The search space is normal-

ized as the center at the original with width A: S

0

≡

{x| |x

1

| ≤ A, |x

2

| ≤ A}. As a preparation, we deﬁne

several objects. The particle α

i

is described by its po-

sition x

i

and velocity v

i

: α

i

= (x

i

,v

i

), x

i

≡ (x

i1

,x

i2

),

v

i

≡ (v

i1

,v

i2

)) and i = 1 ∼ N. The position x

i

is a

potential solution. The personal best of the i-th par-

ticle, pbest

i

= F(x

pbest

i

), is the best of F(x

i

) in the

past history, The local best, lbest

i

= F(x

lbest

i

), is the

best of the personal best pbest

i

for the i-th particle and

its neighbors. For example, the neighbor means the i-

th and both sides particles in the ring topology. The

global best gbest is the best of the personal bests and

is the solution of the present state. In the complete

263

Kubota M. and Saito T..

NESTING DISCRETE PARTICLE SWARM OPTIMIZERS FOR MULTI-SOLUTION PROBLEMS.

DOI: 10.5220/0003623102630266

In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2011), pages 263-266

ISBN: 978-989-8425-83-6

Copyright

c

2011 SCITEPRESS (Science and Technology Publications, Lda.)

graph, the lbest

i

is consistent with gbest.

First, the search space S

0

is discretized into m

1

×

m

1

lattice points as shown in Fig. 1 (a): L

0

≡

{0

11

,l

02

,··· , l

0m

1

}, l

0k

= −A+ kd − d/2, d = 2A/m

1

where k = 1 ∼ m

1

, F(x) is sampled on the discretized

search space D

0

. The target is the AS1 deﬁned by

F(p) < C

1

, p ≡ (p

1

, p

2

) ∈ D

0

(1)

whereC

1

is the criterion. The AS1 is used to make the

LSRs each of which includes either solution. Let t

1

denote the discrete time in this stage and let k

1

denote

the counter of AS1.

Step 1. Let t

1

= 1 and let k

1

= 0. Let N

1

particles

form ring topology. The particle positions x

i

, i = 1 ∼

N, are assigned on D

0

randomly following uniform

distribution on D

0

(Fig. 1 (a)). v

i

, pbest

i

and lbest

i

are all initialized.

Step 2. The position and velocity are updated.

v

i

(t

1

+ 1) = ωv

i

(t

1

) + r

1

(x

pbest

i

− x

i

(t

1

))

+ r

2

(x

lbest

i

− x

i

(t

1

))

x

i

(t

1

+ 1) = x

i

(t

1

) + v

i

(t

1

+ 1)

(2)

where r

1

and r

2

are random parameters in [0,c

1

] and

[0,c

2

], respectively. If x

i

(t

1

+ 1) exceeds D

0

then it

is re-assigned into D

0

. The parameters ω, r

1

, r

2

, c

1

and c

2

are selected from lattice points to satisfy the

condition x(t

1

)inD

0

.

Step 3. The personal and local bests are updated:

x

pbesti

= x

i

(t

1

) if F

1

(x

i

(t

1

)) < F

1

(x

pbest

i

)

x

lbest

i

= x

pbest

i

if F

1

(x

pbest

i

) < F

1

(x

lbest

i

)

(3)

Step 4. If F(x

i

) < C

1

for some i (Fig. 1(b)) then x

i

is declared as the k

1

-th AS1 and the counter number

is increased: p

k

1

= x

i

and k

1

= k

1

+ 1. The position

x

i

is declared as a tabu lattice point and is prohibited

to revisit. x

i

is reset to a lattice point (Fig. 1(c)). v

i

,

pbest

i

and lbest

i

are all reset.

Step 5. Let t

1

= t

1

+ 1, go to Step 2 and repeat until

the maximum time step t

max1

.

In order to make the LSRs each of which is desired

to include the target solution, we select top K of the

AS1s p

1

,··· , p

K

such that F(p

1

) ≤,··· , ≤ F(p

K

) ≤

C

1

. (If k

1

< K then the top k

1

is used). Using the

AS1s, we construct subsets (LSR candidates) succes-

sively for i = 1 to K: the i-th subset S

i

is centered at

p

1

with area (2m

a

d)

2

: S

i

= {x | |x

1

− p

i

1

| < a

1

, |x

2

−

p

i

2

| < a

1

}, a

1

= m

a

d, i = 1 ∼ K where p

i

= (p

i

1

, p

i

2

) is

the i-th AS1 and m

a

is an integer smaller sufﬁciently

than m

1

. The m

a

determines area of S

i

in square

shape. Although this paper uses square-shaped sub-

spaces for simplicity, a variety of shapes should be

tried depending on objectiveproblems. If two or more

subsets overlaps then the subset centered at the small-

est AS1 is survivedand the other subsets are removed.

2

x

(a) t

1

=0

Sol1

Sol2

(b)

1

x

(c)

(d) t

1

=t

max1

2

x

1

x

(e) t

2

=0

(f)

Sol2

2

x

(a) t

1

=0

Sol1

Sol2

(b)(b)

1

x

(c)(c)

(d) t

1

=t

max1

2

x

1

x

(e) t

2

=0

1

x

(e) t

2

=0

(f)(f)

Sol2

Figure 1: Particles movement for two solutions. (a) Initial-

ization, (b) the ﬁrst AS1, (c) position reset, (d) two LSRs.

(e) initialization for local search, (f) AS2.

If we obtain more than M survived subsets, we select

subsets centered at top M of AS1s. We then obtain M

pieces of LSRs and reassign the notation p

i

≡ (p

1

1

, p

i

2

)

to the AS1 in the i-th LSR. If the LSRs include the tar-

get solutions x

i

s

, i = 1 ∼ M then the global search is

said to be successful.

Next, we discretize each LSR onto m

2

× m

2

lat-

tice points: D

i

= {x | x

1

− p

i

1

∈ L

i

, x

2

− p

i

2

∈ L

i

},

L

i

= {l

i1

,··· , l

im

2

}, l

ik

= −(m

a

+ 1/2)d + kd

1

, d

1

≡

2m

a

d/m

2

This D

i

is the i-th discretized LSR. F(x) is

sampled on D

i

. The target is the AS2 deﬁned by

F(q

i

) < C

2

, q

i

≡ (q

i

1

,q

i

2

) ∈ D

i

(4)

where C

2

(< C

1

) is the criterion. The local search

operates in parallel in D

i

(Fig. 1 (e) (f)).

Step 1. Let t

2

denote the discrete time and let t

2

= 1.

Positions of N

2

particles in complete graph are as-

signed randomly on D

i

(Fig. 1 (e)). v

i

, pbest

i

and

gbest are all initialized.

Step 2. The position and velocity are updated.

v

i

(t

2

+ 1) = ωv

i

(t

2

) + r

1

(x

pbest

i

− x

i

(t

2

))

+ r

2

(x

gbest

− x

i

(t

2

))

x

i

(t

2

+ 1) = x

i

(t

2

) + v

i

(t

2

+ 1)

(5)

where r

1

and r

2

are random parameter in [0,c

1

] and

[0,c

2

], respectively. If x

i

exceeds D

i

then it is re-

assigned into D

i

. The parameters ω, c

1

and c

2

are

selected to satisfy x

i

∈ D

i

.

Step 3. The personal and global best are updated:

x

pbesti

= x

i

(t

2

) if F

1

(x

i

(t

2

)) < F

1

(x

pbest

i

)

x

gbest

= x

pbest

i

if F

1

(x

pbest

i

) < F

1

(x

gbest

)

(6)

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

264

Step 4. If F(x

gbest

) < C

2

for some i then we obtain

one AS2 and the algorithm is terminated.

Step 5. Let t

2

= t

2

+ 1, go to Step 2 and repeat until

the maximum time step t

max2

.

If the object is the AS2 only, the particles are of-

ten trapped into either solution or local minima and

are hard to solve the MSP. Our NDPSO tries to sup-

press the trapping by global search for the AS1 with

discretization (sampling) of the objective function.

3 NUMERICAL EXPERIMENTS

In order to investigate the algorithm capability, we

have performed basic numerical experiments for the

Himmelblau function with four solutions as illus-

trated in Fig. 2 (a):

F

H

(x) = (x

2

1

+ x

2

− 11)

2

+ (x

1

+ x

2

2

− 7)

2

min(F

H

(x

i

s

)) = 0, i = 1 ∼ 4,

(7)

x

1

s

= (−2.805118,3.131312) ≡ Sol1,

x

2

s

= (3,2) ≡ Sol2,

x

3

s

= (−3.779310,−3.283185) ≡ Sol3,

x

4

s

= (3.584428,−1.848126) ≡ Sol4

S

0

= {x | |x

1

| ≤ 6, |x

2

| ≤ 6 }.

We have selected m

1

, C

1

and t

max1

as control param-

eters and other parameters are ﬁxed after trial-and-

errors: N

1

= N

2

= 20, ω = 0.7, c

1

= c

2

= 1.4, K = 30,

m

1

/(2m

a

) = 8, m

2

= 32, t

max2

= 50 and C

2

= 0.04.

Fig. 2 (b) to (f) and Fig. 3 show a typical example

of the global search process for m

1

= 64, C

1

= 5 and

t

max1

= 50. The NDPSO ﬁnd the ﬁrst AS1 at t = 4

and ﬁnd the other AS1s successively. At time limit

t

max1

, the NDPSO can construct all the four LSRs suc-

cessfully. Each LSR has 8

2

(8 = 2m

a

= m

1

/8) lattice

points. Fig. 2 (g), (h) and Fig. 4 show the local search

process where the NDPSO can ﬁnd all the approxi-

mate solutions.

We evaluate the global search by success rate (SR)

that means rate of ﬁnding all the LSRs in 100 trials for

different initial states. Table 1 shows the SR of global

search for m

1

and C

1

. For m

1

= 32, 64 and 128, the

LSR has 4

2

, 8

2

and 16

2

lattice points, respectively in

the global search. The LSR is divided into m

2

× m

2

lattice points for the local search. We can see that

C

1

is important for ﬁnding LSRs. As C

1

increases

the SR increases and tends to saturate. For smaller

C

1

, the NDPSO operates like standard analog PSO in

principle and tends to trap local solution/minimum.

For larger C

1

, the NDPSO has possibility to ﬁnd a

suitable AS1 before the trapping. As m

1

increases,

the resolution becomes higher and the SR tends to in-

crease; however, the computation cost also increases

(c) t

1

=4

2

x

1

x

6−

6−

6

6

(f) t

1

=t

max1

AS1

Sol1

Sol2

Sol3

Sol4

(a)

(b) t

1

=0

(d) t1=7

(e) t

1

=9

(g) t

2

=0

(h) t2=4

2

x

1

x

Sol3

AS1

AS1

(c) t

1

=4

2

x

1

x

6−

6−

6

6

(f) t

1

=t

max1

AS1

Sol1

Sol2

Sol3

Sol4

(a)

(b) t

1

=0

(d) t1=7

(e) t

1

=9

(g) t

2

=0

(h) t2=4

2

x

1

x

Sol3

AS1

AS1

Figure 2: Particles movement of F

H

. (a) contour map, (b)

initialization, (c) to (e) global search process, (f) four LSRs

(g) initialization for local search, (h) AS2.

1

t

1

C

0

0

10

20

40

1max

t

))((

i

pbestH

xF

mim

1

t

1

C

0

0

10

20

40

1max

t

))((

i

pbestH

xF

mim

Figure 3: Global search process of F

H

. mim(F

H

(x

i

)) means

the minimum of F

H

at time t

1

.

of course: there exists a trade-off between the SR and

computation cost.

We evaluate the local search by the SR and the

average number of iteration (#ITE) of ﬁnding all the

NESTING DISCRETE PARTICLE SWARM OPTIMIZERS FOR MULTI-SOLUTION PROBLEMS

265

)(

gbestH

xF

2

t

2

10

−

0

1.0

1

4

8

10

2

C

1

D

2

D

3

D

4

D

)(

gbestH

xF

2

t

2

10

−

0

1.0

1

4

8

10

2

C

1

D

2

D

3

D

4

D

Figure 4: Local search process of F

H

.

AS2s in 100 trials for different initial states. Table 2

shows the SR/#ITE of local search after the successful

global search. We can see that the NDPSO can ﬁnd

all the AS2 speedily.

Table 1: SR of global search of F

H

for t

max1

= 50.

C

1

= 3 C

1

= 5 C

1

= 10 C

1

= 20 C

1

= 30

m

1

= 32 20 36 59 79 84

m

1

= 64 32 53 74 92 94

m

1

= 128 46 61 80 91 95

Table 2: SR/#ITE of local search of F

H

for t

max1

= 50.

C

1

= 3 C

1

= 5 C

1

= 10 C

1

= 20 C

1

= 30

m

1

= 32 100/5.30 100/5.60 100/5.46 100/5.55 100/5.57

m

1

= 64 100/5.01 100/5.34 100/5.41 100/5.43 100/5.49

m

1

= 128 100/5.08 100/5.20 100/5.36 100/5.42 100/5.55

Table 3: SR of global search of F

H

for C

1

= 5.

m

1

= 32 m

1

= 64 m

1

= 128

t

max1

= 10 8 8 8

t

max1

= 30 31 40 47

t

max1

= 50 36 53 61

t

max1

= 100 - 57 70

t

max1

= 200 - 60 71

t

max1

= 400 - - 71

t

max1

= 800 - - 72

Table 3 showsthe SR in global search for t

max1

and

m

1

. The SR increases as t

max1

increases. For m

1

= 64

and 128, the SR saturates and t

max1

= 50 (or 100) is

sufﬁcient for reasonable results. The parameter t

max1

can control the SR and computation costs.

4 CONCLUSIONS

The NDPSO is presented and its capability is investi-

gated in this paper. Basic numerical experiments are

performed and the results suggest the following.

1. The parameters m

1

and C

1

can control rough-

ness in the global search that is important to ﬁnd all

the LSRs successfully. Higher resolution encourages

trapping and suitable roughness seems to exist.

2. Parallel processing of the local search in LSRs

is basic for efﬁcient search. If LSRs can be con-

structed, the AS2s can be found speedily and steadily.

3. The discretization is basic to realize reliable

and robust search in both software and hardware.

Future problems are many, including analysis

of search process, analysis of role of parameters,

comparison with various PSOs (Engelbrecht, 2005)

(Miyagawa and Saito, 2009) and application to prac-

tical problems (Valle et al., 2008) (Kawamura and

Saito, 2010).

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