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Parallel Simulated Annealing with Adaptive Neighborhood using Different Target Acceptance Ratio

Hiroki H IRAO * , Yuichiro U EDA * , Mitsunori M IKI ** and Tomoyuki H IROYASU ***

(Received January 20, 2009)

This paper deals with a new approach in Simulated Annealing (SA), and proposes an adaptive neighborhood mechanism for continuous optimization problems. When applying SA to continuous optimizing problems with nu- merous local optima, the automatic control of the size of the neighborhood becomes necessary to obtain good per- formance.We have already proposed the method called Simulated Annealing with Advanced Adaptive Neighborhood (SA/AAN).This method has an adaptive neighborhood adjustment mechanism maintaining a given target acceptance ratio, and it shows very good performance for continuous optimization problems.The target acceptance ratio in this method is determined experimentally. In order to overcome this problem, SA/AAN is perfomed in parallel with differnt target acceptance ratios.The proposed methods are applied to solve many continuous optimization problems, and it is found that the methods are very useful and effective.

Key words optimization, simulated annealing, parallel, adaptive neighborhood

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* Graduate Student, Department of Knowledge Engineering and Computer Sciences, Doshisha University, Kyoto Telephone:+81-774-65-6921, Fax:+81-774-65-6716, E-mail:[email protected]

** Department of Knowledge Engineering and Computer Sciences, Doshisha University, Kyoto Telephone:+81-774-65-6930, Fax:+81-774-65-6716, E-mail:[email protected]

*** Faculty of Life and Medical Sciences, Doshisha University, Kyoto

Telephone:+81-774-65-6932, Fax:+81-774-65-6019, E-mail:[email protected]

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$%³ E ®

– H

9;

‘

±

Ä

š Ê

‘-c p p ‹

‘

d+=

¨ S š Ê {

‘ V Ï´puo

Ö

q

v



v # ‘

Ä

š Ê ‘ ÅÆ

º A v

‹1

‘i Ö

tƒo

ˆ ‘

­ ‹ F n ¾ ¿

{ ‚ à 4 Ó ˜  v  D Ö ˆ

(*U

i ”

$%b&'²ˆ

­²‹

§ƒ¨

¾¿

‚à v 

Ö t

‘

± “ ˆ

u Ä

š Ê

‘ c p

Ã

tms $%

³m˜

+ 2

‘ D

¹Úp

¼½

#¾¿

‚ - Ö q ˆ > »

Ä

“ ‚ ­ ¨ t

$

ºt v 

7.

7.1

!

"$#

 Ž q < =  Ž { À Á

‚&%$'

p ‹ e f ‚

Fig.

7

‘ 

Fig. 7

i ” Ù

# ‘ ] »

Ï(u

{

”Ž

ˆ

Table 1. Parameters in each method.

Function Rastirigin Griewank

Total steps 25600 × 4 76800 × 4

Cooling steps 32

Max.(Initial) temperature 10 20

Min.(Final) temperature 0.01 0.001

Initial neighborhood range 1.0 5.12 6.0 512

Neighborhood adjustment interval 50

Neighborhood range’s parameter

adjustment interval 200

Optimum solution area Optimum solution area

10-4 10-2 100

Energy

Trial number (sorted)

10-6 10-4 10-2

Energy

Trial number (sorted)

(a) Rastrigin function (b) Griewank function

0 20 40 60 80 100

㩷㪧㪪㪘㪆㪝㪥 㩷㪧㪪㪘㪆㪘㪘㪥 㩷㪘㪩㪧㪪㪘

0 20 40 60 80 100

㩷㪧㪪㪘㪆㪝㪥 㩷㪧㪪㪘㪆㪘㪘㪥 㩷㪘㪩㪧㪪㪘

Fig. 7. Distribution of optimum solutions in 3 methods.

100

- S ã - »

²

‚)

s*

{uˆ

 ±

‘

[^\

‘F - z #

‚XE

p Ï » v 

n ]

”•–

—

i

Table 1

‘ 

 

 

ÕnO

Ñ

$%³

‚

» v

!

SA : PSA/FN

ÕnÝÞß

‚àá

q

Ãv

!

SA : PSA/AAN

!n

v

ÝÞß

‚àá

q

Ãv

!

SA : ARPSA

o ‹

PSA/FN

‘ ] ` v O Ñ $%²³ i+, N ƒ‘

Î Ï q

1·‹²²Õ nb

‚b

» Ï » v  D pϲ

PSA/AAN

{ à·á

Ý Þ ß i

"-#

q º·t Ï » v

0.1

q p

3)

ARPSA

{ à á Ý Þ ß i

0.05

0.1

0.2

0.4

{

4

ã

{

qp‹



Fig. 7

± Ä “ ‘ ã » Ï i » S t { Ù

# ‘

] » Ï ²

ARPSA

˜ /Úne3f ‚ ­ Ï » v 

o ‹ Ä š Ê ª { c p ß i

(a)Rastrigin

Ù #

ˆ i

PSA/FN

‘. v { {

(b)Griewank

Ù

# ˆ

i

PSA/FN

q(&/ {1> » e f ‚ ­ Ï » v  ‹ K

PSA/FN

{ +$, N ‘ § § v „ …10 ' 6 ‚ 2 à v

q

< =

Žm˜

ˆmv

q:

¨ t v  ΠÏ

<

=

Ž ˜

"3#

²Ž

±

ƒne/f

‚

­²Ï

» v q v 

Fig. 1. Algorithm of Simulated Annealing.
Fig. 2. Effect of the neighborhood range on the en- en-ergy. 3.2 €  $%³m˜ ¾¿ ‘  v  qp Ïk$%³´q Ä “ { ÙXŽ ‚ Fig
Fig. 4. Concept of ARPSA.
Fig. 5. History of the neighborhood range with dif- dif-fernt target acceptance ratio.
+2

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