The 19th Annual Conference of the Japanese Society for Artificial Intelligence, 2005           CSP      1A1-01  "!
#%$ &(')+*(,.-0/+1(/+1(2 354.6+,7158+9(:Lagrange Neural Network for Solving CSP Which Has Objective Function

Takahiro Nakano

Masahiro Nagamatu

Graduate School of Life Science and Systems Engineering, Kyushu Institute of Technology

We proposed a Lagrange Neural Network called LPPH-CSP to solve the CSP. This method is never trapped byany point which is not a solution of the CSP. From experimental results of the LPPH-CSP, we confirmed that ourmethod is as efficient as the GENET which is a famous CSP solver. In addition, unlike other conventional CSPsolver, our method is a continuous-valued method and it can update all variables simultaneously. Therefore, we
can expect the speed-up of the LPPH-CSP if it is implemented by the hardware such as FPGA. In this paper, weextend the LPPH-CSP to treat the linear inequality constraints. By using this type of constraint, we can represent
various practical problems more briefly. In this paper, we also define the CSP which has an objective function, andwe extend the LPPH-CSP to solve this problem. In the experiment, we apply our method and the OPBDP to theWLP and compare the effectiveness.

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D7$8(E(F2GHIm­·P • X = {X1 , X2 , · · · , Xn }

1.

CSP)

• D = {D1 , D2 , · · · , Dn }
Di

LPPH-

CSP[Nakano 2004]

CSP

LPPH-CSP
GENET[Davenport 1994]
CSP
LPPH-CSP

LPPH-CSP

LPPH-CSP

0Xi

Di

j
VVP

(Variable-Value Pair)
CSP
ALT(n, S)[at-least-n-true constraint] ALF(n, S)[at-leastn-false constraint] AMT(n, S)[at-most-n-true constraint]
AMF(n, S)[at-most-n-false constraint]
SVVP
ALT(n, S)SnVVP

11CSP

CSPX

crijσcrij xij ≤ σ.

VVPxijCrÁcÂc¢+L P Vc·PlÅc¢  ¨ D  WqÔ\q^dT©¡fS
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3.

CSP

LPPH-CSP
warehouse location problem(WLP)[Scaparra

2001]

CSP

3Xxij”

LPPH-CSP

OCSP)

.

C

CSP

VLSI

2Xi

• C = {C1 , C2 , · · · , Cm }

CSP

0.

(X, D, C)T

Lagrange Neural Network for CSP
Xidxijdt=dwr
dt=dWrdt=Dijxij
LPPH-CSP

xij (1 − xij )

mX

xij

0

1Wr wr srij (x), for all VVP xij ,

r=1

hr (x) − αwr , r = 1, 2, · · · , m,(PWr hr (x), if Cr = ( crij xij ≤ σ),0,otherwise.

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A:srij (x)wr

808-0135
2-4 tel/093-695-6088
fax/093-695-6088 nagamatu@brain.kyutech.ac.jp

1CrCrxij

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The 19th Annual Conference of the Japanese Society for Artificial Intelligence, 2005

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LPPH-OCSP

Wr
LPPH-CSP

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hr (x) srij (x)Cr =ALT(n, S)

3.1

hr (x)

ALT

srij (x)1hr (x) = 1 − NMax(n, S),(1 − NMax(n + 1, S), if xij ≥ NMax(n, S),srij (x) =hr (x),otherwise,.

(PAMF

2600

crij xij ≤ σ)hr (x)

srij (x)Pifcrij xij − σ < 0,P 1( crij xij − σ) , otherwise,|crij |µ ³³´¶´
1[ij,1][ij,0]
P.
− hr xhr xsrij (x) =|crij |³³´´
hr x[ij,0]hr x[ij,1]xij
0 1hr (x) =0,P

OBPDP[Barth 1995]
2 30
15

LPPH-CSP

best know value of objective function

Cr = (3.2

WLP

AMT
WLP 0-1
0-1
LPPH-CSP
WLP
CPU

OPBDP

hF G  FcG  FcG SÕ{fE_mPEí[à|îhµ FcGô¹®Plé|ê]Ð]¿®ã[·W"+!fm^]¡fS¢a¦®­[^l}¡\Æ^w<P & jP FaG P S D ÈdP®ã Ô¿$&! ¬AMT

LPPH-OCSP
WLP
ALT

3

where NMax(n, S) = nth maximum value in S.

ALF

D î"Â

5.

CSP

2400

best know value of objective function

LPPH-CSP

F (x)

OPBDP
LPPH-OCSP_init1
LPPH-OCSP_init2
LPPH-OCSP_init3

2400
2200
2000
1800
1600
1400
1200
1000
05

10

15

2025time (sec)

30

OPBDP
LPPH-OCSP_init1
LPPH-OCSP_init2
LPPH-OCSP_init3

2200

2000

1800

1600

1400

1200

1000

800

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YUW
N vÕÈ|P|ã<ö¿$!]f^·} Dãj ¨ThSí]VþcUh¦wì+hx Q ydUwUEW])h{fR®^cva¡|ãmSEv·¨©¼w½®äPYUw¯[òa}XWîhYfµ LlP7ZM ¿'aSEò[UYWîEÂdv]ìjã9 µ^]¡fSÔ¢[\Ô \Æ^l}
1: problem1

2: problem2rhr (x)

xij

CSP

LPPH-CSP

1CSP

xij

CSP

LPPH-OCSP

4.1

OCSP

(OCSP)CSP OCSP)
find x,such that minimize] ^`_baE(x),subject to {Cr |r = 1, 2, .., m}.

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4.2

Lagrange Neural Network for OCSP
OCSP

[Nakano 2004] T.Nakano, M.Nagamatu: Solving CSP via
Neural Network Which Can Update All Neurons Simultaneously, In Proceedings of the SCIS & ISIS, 2004.

LPPH-CSP
OCSP
wahouse location problem(WLP)
E(x)¶µX∂F (x)
dxij,= xij (1 − xij )Wr wr xij − LHij δdt
∂xij
dwr
dt

=dWr
dt

=dHij
dt

=xij

=(dLdt

LPPH-CSPOPBDP[Barth 1995]
WLP
OPBDP

[Davenport 1994] A.Davenport, E.Tsang, C.j.Wang, andK.Zhu: GENET: A Connectionist Architecture forSolving Constraint Satisfaction Problems by Iterative
Improvement, National Conference on Artificial Intelligence, pp.325-330, 1994.

hr (x) − αwr ,(PWr hr (x), if Cr = ( crij xij ≤ σ),0,otherwise,[Scaparra 2001] M.P.Scaparra, M.G.Scutella: Facilities,Locations, Customers: Building Blocks of Location
Models. A Survey, TR-01-18, 2001.

∂F (x)− βHij ,
∂xij

η,−²,

[Barth 1995] P.Barth: A Davis-Putnam Based Enumeration Algorithm for Linear Pseudo-Boolean Optimization, Tech. Rep. MPI-I-95-2-003, 1995.

if all constraint are satisfied,otherwise.

2