Estimation of Distribution Algorithm for Solving Linear Bilevel Programming Problems

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S Qingha iNormalUniverstiy,Xining810008,China

r o h t u a g n i d n o p s e r r o C *

: s d r o w y e

K Linear blieve lprogramming problems , Esitmaiton of distribuiton algortihm, Opitma l ,

s n o it u l o

s Probablitiydistribuiton.

.t c a r t s b

A Int hismanuscript ,anewapproach ,estimationofdistributionalgorithm(EDA),i sutilized e

n i l e h t e v l o s o

t arbileve lprogrammingproblem .Newi ndividualsaresampled fromt heprobability .

w o n o t p u d e n i a t b o n o i t u b i r t s i

d Some tested problemsaresolved by thepresented EDA and the .

m h t i r o g l a d e s o p o r p e h t f o y t i l i b i s a e f d n a y c n e i c i f f e e h t w o h s s t l u s e r n o i t a l u m i s

n o it c u d o r t n I

b e h

T ileve lprogrammingproblem( BLPP)i sahierarchica loptimizationproblemconsistingoft wo s

m e l b o r p n o i t a z i m i t p

o a tdifferentl evels .Thet wooptimizationproblemsarecalledt heleader’s and r

e w o l l o f e h

t ’ problems ,respectively, also termed the upper leve lproblem (UP) and lower leve l m

e l b o r

p (LP). Thiskindofproblemisfirstlyproposed b y VonStackelberg[1] ,whichi sformulated s

w o l l o f s a y l l a c i t a m e h t a m

0

0

X x

X y

) y , x ( F n i m

) y , x ( G . t. s ) P B

( _{min (fx ,y)}

) y , x ( g . t. s

∈

∈

 

≤ 

 

 _≤



( 1)

where _x∈X∈Rn_{, y∈Y∈R}m _{are called decision variables of the UP a end t Ph L ,respectively .} m

n _{R R}

R : f ,

F × → ,_{G :R}n_×Rm_→Rp 

�_{g :R}n_×Rm_→Rq_.

c i m o n o c e g n i v l o v n i , e f i l l a e r n i s n o i t a c i l p p a f o d l e i f e d i w a s a h P P L

B [2] ,structura ldesign[3] ,

n o i t a t r o p s n a r t n a b r

u [4],t rafficassignment[5] ,pollutioncontrol[6],etc .BLPPs ea r alwaysdivided n

o N d n a ) P P L B L ( P P L B r a e n i l : s e i r o g e t a c o w t o t n

i -LinearBLPP(NLBLPP).

e m e r t x e e m o s t a r u c c o n a c n o i t u l o s l a m i t p o l a b o l g s t i d n a s P P L B t s e l p m i s e h t f o e n o s i P P L B L

n o i g e r t n i a r t s n o c e h t f o t n i o

p ][7 .Even so ,i tisstil lastrongly NP-hardproblem [ ]8 .Based on the l

a i c e p

s characteristicsoft heLBLPP ,someresearchershaveproposedafewefficien talgorithms ,for s

d o h t e m t n i o p e m e r t x e , e c n a t s n

i ][ , 7 Kth-bes t[7,9] ,Kuhn-Tucke rmethod [10] ,branch and bound s

e h c a o r p p

a [11], penaltyfunctionmethods[12] ,geneticalgorithm[13] ,etc. t

a v i t o

M ed by [13] ,considering tha tthe computationa l cos tof solving linear programming y t i l a m i t p o e h t t p o d a h c i h w , A D E l e v o n a p o l e v e d e w , n o i t u l o s e l b i s a e f a g n i n i a t b o n i s m e l b o r p

m m a r g o r p r a e n i l f o s m e r o e h

t ingandsearchoptima lbasei nafinitespace .Int hismanuscript ,EDAi s e c n a m r o f r e p r e t t e b e h t s e t a c i d n i A D E : t c a f e h t n o d e s a b s i h c i h w , e m a r f m h t i r o g l a n i a m a s a d e t p o d a

r e h t o o t d e r a p m o c n e h

w heuristicmethods in solving optimization problems with finite search .

s e c a p s

n i m e l b o r p d e s s u c s i d e h t t n e s e r p e W . s w o l l o f s a d e z i n a g r o s i r e p a p s i h

T Section2 .InSection3 ,

n o i t u b i r t s i d f o n o i t a m i t s

e algorithmisbrieflydescribed .Weprovide thedesignoft healgortihmbased A

D E n

d e s s u c si

D Problem

o m e s o h w P P L B L r e d i s n o c t s u j e w , r e p a p s i h t n

I de lcanbedescribedasfollows.

1 1

1 1 1

2

2 2 2

0 0

T T X

x

T Y

y

y d + x c = ) y , x ( F n i m ) P U (

b y B + x A . t. s

,s e v l o s y e r e h w )

P B L (

y d = ) y , x (f n i m ) P L (

b y B + x A . t. s

. y , x

∈

∈

 

≤ 

   

 _≤



 _{≥ ≥}



( 2)

where _{x ,c1∈R}n _,

2 1 ,d Rm d

,

y ∈ ,_b1∈Rp _, 2 Rq

b ∈ ,_A1∈Rp×n _,

1 Rp m B _∈ × _,

2 Rq n A _∈ × _,

2 Rqm B _∈ × _.

s i ) 2 ( t a h t e r u s n e o

T wel lposed ,we always assume : (i) The common constraint region S is ;

t c a p m o c d n a y t p m e n o

n ( ii)Foral ldecisions taken by the leader ,the followerhas some room to ;

d n o p s e

r ( iii)Thefollower hasuniquey(x)foreach fixed x∈S

( )

X ,theprojection of S onto the e

c a p s n o i s i c e d s ’ r e p p

u ; ( vi )TherankofmatrixB2 si q;

u c s i d s

A ssedi n[1 ] ,4 thefollowerof(2)canbet ransformedas

yin (fx ,y)=dy m

y , ) x ( b = y B . t. s  

 ≥



    _

 0 ( 3) e

r e h

w B=

(

B₂,I _q

)

,b

( )

x =b₂-A₂x ,d=

(

d₂ ,0

)

,y=

(

y ,y₀

)

∈Rm+q, y₀ isaslackvectorof qdimensions. r

o

F a yn base B of(3) ,wecanj udget hefeasibilityandoptimalityoft hebaseaccordingt o )( . 4

1

1

-B

0 ) x ( b B

0 B B d -d

 ≥



 _≥

 

) ( 4

wherethevaluesof d_B arecomponentsofd corresponding to thebase .Ifthereexists a tleas tan

( )

X S

x∈ satisfying )(4 ,asinglel evell inearprogrammingisderivedasfollows

x

-) x ( y d + x c n i m

b ) x ( y B + x A . t. s

) x ( b B

x 



 ≤



 ≥

 _≥





 

1 1

1 1

1

1 ₀

0

)( 5

where y(x) comes from y with basic components _B-1_{b( )x} and non-basic components 0’s .The

o t g n i d n o p s e r r o c l a u d i v i d n i e h t f o s s e n t i f e h t s a n e k a t e b n a c ) 5 ( f o e u l a v e v i t c e j b o l a m i t p

o B.

t s

E ima itono fDsitribu itono fAlgortihm

m h t i r o g l a f o n o i t u b i r t s i d f o n o i t a m i t s

E (EDA) ,firs tintroducedby MuhlenbeinandPaaβ [15 ,] si a r

e h t i e n e r a e r e h t , s A G e k i l n U . s m h t i r o g l a y r a n o i t u l o v e f o s s a l c w e

n crossovernormutationoperators

a g n i t c u r t s n o c h g u o r h t n o i t a l u p o p e h t f o n o i t a m r o f n i l a c i t s i t a t s l a b o l g s e r u t p a c A D E , d a e t s n I . A D E n i

s e t a d i d n a c w e n s e t a r e n e g d n a r a f o s d e n i a t b o s n o i t u l o s r o i r e p u s d e t c e l e s e h t r o f l e d o m c i t s i l i b a b o r p

i t u b i r t s i d c i t s i l i b a b o r p d e h s i l b a t s e e h t g n i l p m a s a i

v on .In general ,EDAendure sthefollowingthree .t

p e t

S 1 Selec tsuperiori ndividualsfromapopulation. p

e t

S 2 Estimatet heprobabilitydistributionfromt hechoseni ndividuals. p

e t

S 3 Generatenewi ndividuals(.ie. ,offspring)fromt heestimateddistribution.

m h ti r o g l A f o n g is e

D BasedonEDA

h c a

E individua li sencodedwithabaseof(3) ,soani nitia lpopulationwitht hesizeofp - eop siz canbe g

n i t c e l e s y l l a c i h p a r g o c i x e l y b d e t a r e n e

g p - eop siz din ividuals. Forany feasiblebase, y(x) islinear y

l n o d e t a i c o s s

a w ithx, da n problem(5)i ssolvable .Iti sr easonablet otaket heoptima lobjectivevalue .l

a u d i v i d n i e h t f o s s e n t i f e h t s a ) 5 ( f o

i e h t e t o n e d e w , y l l a r e n e

G ndicesoft heconstrain tmatrixby( ,1 ,2 ,… m+q),i nwhichqcolumnsare g

n i n i a t b o , l a u d i v i d n i n a s a d e t c e l e

s qstochasticvariables X1 ,X2 , ,Xq ,where Xi rangesfrom1t o

q +

m .According to the scale N ,i tis easy to ge tthe corresponding probability distribution .New y t i l i b a b o r p e h t e t a d p u n a c e w , t a h t r e t f a , n o i t u b i r t s i d t s a l e h t m o r f d e t a r e n e g e r a s l a u d i v i d n i

distributioni nt hesimilarway.

o t y l e k i l y r e v s i e r e h t t a h t y h t r o w e t o n s i t a h

W choose the same column during creating new

. m e h t e c a l p e r o t n e s o h c t o n s n m u l o c r e h t o e k a t d n a s n m u l o c d e t a e p e r e h t e t e l e d e w , o s f i , s l a u d i v i d n i

y b e m e h c s n o i t a b r u t r e p a e d i v o r p e w , n o i t a l u p o p w e n n i s n o i t u l o s f o y t i s r e v i d e h t t a h t g n i r e d i s n o C

t g n i s a e r c n

i heselectionprobabilityofcolumnsi ft heyare0sanddecreasingothervaluesofselection .

n o i t a r b i l i u q e f o e k a s e h t r o f y t i l i b a b o r p

d e s o p o r

P Algortihm

. s w o l l o f s a n e v i g e r a m h t i r o g l a e h t f o s l i a t e d e h T . d e t n e s e r p s i P P L B L r o f A D E n a , n o i t c e s s i h t n I

t

S pe 1 I(niitailzaiton) Firstly ,an initialization population pop( 0) with the size fo pp - eo siz is f

o g n i t s i s n o c , d e t a r e n e

g B( )j0=

(

i₁,i ₂ , ,i _q

)

,j=1 , ,pop-szie,i _k ,k=1 , ,q.

−

 

 are indices fo columns of the

d e t c e l e

s base, tl e t=0 .Evaluatet hefitnessofeachi ndividuali npop(t). p

e t

S 2( Seleciton)Sortt he ppo (t) accordingt ot hef itnessvaluesandselec tN“better”i ndividualsas O1 bytheroulettewhee lselection.

p e t

S 3 (Create probab litiy distribuiton) Compute the probability distribution according to n

i s l a u d i v i d n

i O1. Duringt hisprocess,i nordert oi mprovet hediversityoft hepopulation ,weassigna

h c i h w , s n m u l o c r e h t o f o s e i t i l i b a b o r p e h t e s a e r c e d e l i h w , d e t c e l e s t o n s n m u l o c e h t r o f y t i l i b a b o r p y n i t

t i l i b a b o r p l a t o t e h t s e k a

m yoft hesecolumnsbe1 . p

e t

S 4(Sample )Ont hebasisofprobabilitydistribution ,generateanewpopulationO2containing

(p -op size-N) individuals. p

e t

S 5(Termina iton crtierion )Evaluate the fitness values of each individua lin O2 ,if the

r c n o i t a n i m r e

t iterion is satisfied , stop , outpu t the optima l solution p_t

( )

B_i ; Otherwise , le t

2

1 O , t=t +1,

O =

O ∪ returnt oStep2.

a r g o r p l e v e l i b r a e n i l e h t h t i w l a e d n a c e w t a h t s i m h t i r o g l a s i h t f o e g a t n a v d a r o j a m

A mmingeven

e l b a i r a v s ’ r e d a e l e h t h g u o h

t nisveryl arge.

Simula iton

e h t f o y t i l a u q e h t f o s m r e t n i d o h t e m r u o f o y c n e i c i f f e e h t w o h s o t m i a e w , s t n e m i r e p x e e s e h t h t i W

[ B B A G , s e r u t a r e t i l e h t n i d e d i v o r p s e h c a o r p p a e h t g n o m a e c n i S . s n o i t u l o

s 13]hasbeeni llustratedt o

h t r o f t n e i c i f f e e

b i skindofproblems ,thusweonlycomparethecomputationa lresultswithGABB . 4

A D M A g n i v a h C P a n o d e m r o f r e p e r e w s t n e m i r e p x e e h

T -3300M APU with Radeon(tm) HD

0 2 R B A L T A M n i n e t t i r w e r e w s e d o c m h t i r o g l a e h T . z H G 9 . 1 s c i h p a r

G 10a.WetakeN astheceiling

f

e h t w o h s o t d e t s e t e r a s e l p m a x e e l p m i s e e r h t , y l t s r i F . s t r a p e e r h t f o d e s o p m o c e r a s t n e m i r e p x E

e p e r a s m e l b o r p e l a c s r e g r a l , y l d n o c e S . m h t i r o g l a d e s o p o r p e h t f o y c n e i c i f f e d n a y t i l i b i s a e

f rformed .

. B B A G d n a A D E e h t f o e c n a m r o f r e p e h t e r a p m o c e w , y l l a n i F

1 e l b a T n i s m e l b o r p d e t s e t l l a r o

F (threesimpleexamples) ,wese tp - eop s iz be3andt healgorithm o

r P t s e T ‘ . 0 1 s e h c a e r n o i t a r e t i e h t n e h w s p o t

s b.’standsforthetested problems ,‘Ind.’meansnew f

o t c u d o r p e h t y b d e n i f e d , n o i t a r e t i h c a e n i d e t a r e n e g s l a u d i v i d n

i pp - eo siz andi terationnumber ,and e

h t o t s r e f e r ’ n u R

‘ run times on each of problems .We also presen tthe average time ,CPU ,of r

e v a e h t d n a s n u r 0 2 n i n o i t u l o s l a m i t p o n a g n i n i a t b

o agevalueoft heearlies tCPUt ime ,ET ,finding f o e u l a v e g a r e v a e h t s e b i r c s e d ’ I E ‘ n m u l o c t s a l e h T . s d n o c e s n i e r a o w t e h t f o h t o b , n o i t u l o s t s e b e h t

n o i t a r e t i e h

t sa twhichtheoptima lsolutionappearsfort hefirstt ime. e

l p m a x

E 1 ][ 7

1

2

2

2

2 1 2

1 2

1

2 1 2

1 x a m

x a m

, 1 2 2

, 6 2 , 0 1 2 . .

. 2 , 1 , 0 , 8 1 2 , 8 3

-x

x

i x

x

x x x

x x

x t s

i x x

x x

x    

 − ≤ − ≤ − ≤

 _{+ ≤ + ≤ ≥ =}

 o

e h

T ptima lsolutionx= (16,11) ,max F = .1 1 e

l p m a x

E 2 [16]

2 1

2 1

2 1 2

1 2

1

2 1 2

1 2

1

2 1 2

1

0 2 2 )

, ( x a m

0 1 )

, ( x a m

, 1 ,

1 ,

3 .

.

, 7 1 0 6 6 1 6 , 7 3 0 6 6 6 1 , 1

. 2 , 1 , 0 , , 7 2 0 6 6 1 6 1 , 7 2 0 6 6 6

-x

y

i y

y x y x F

y y y x f

y y x y y x y y x t s

y y x y

y x y

y x

i y x y

y x y

y x

− + = 



+ = 

 _{+ + ≤ + − ≥ + + ≤}



 _{− + ≤ − + ≤ − + ≤}



= ≥ ≤

+ − ≤

+ −   o

e h

T ptima lsolutionx = (1.75;1,0.25) ,max F = 4.75. e

l p m a x

E 3 1[ 1]

3 2 1 2 1

3 2 1 2 1

3 2 1 1 3 2 1

3 2 1 2

4 0 4 4 4 8 n i m

2 n

i m

, 1 5 0 2 2

1 2 .

.

. 3 , 2 , 1 2 1 0 1

5 0 2

2 _{i j}

y y y x x

y y y x x

y y y x y y y t s

j i

y x y y

y x

− − + − 

 _{+ + + +}



 _{+ + ≤ − + + ≤}



 _{+ − − ≤ ≥ = =}



-. ,

, , , , , .

e h

T optimalsolutionx = (0,0.9,0,0.6,0.4) ,minF = -29.2.

1 e l b a

T . Computationa lresultson examples1-3. o

r P t s e

T b. Ind . Run Solutions CPU(s) ET(s) EI

e l p m a x

E 1 3 0 2 0 Best: X=(16,11) F=11. ) 1 1 , 6 1 ( = X : t s r o

W F=11.

7 2 .

0 0 6 .0 3 7 .

e l p m a x

E 2 3 0 2 0 Best :X=(1.75;1,0.25) F = 4.75 ) 5 2 . 0 , 1 ; 5 7 . 1 ( = X : t s r o

W F=4.75

.

0 73 0 6.0 3 .5

e l p m a x

E 3 3 0 2 0 Best: X=(0,0.9,0,0.6,0.4)

F = -29.2 ) 4 . 0 , 6 . 0 , 0 , 9 . 0 , 0 ( = X : t s r o W

F = -29.2

9 2 .

1 1.17 5 0.

. s e r u t a r e t i l n i d e d i v o r p e s o h t s a s n o i t u l o s l a m i t p o e m a s e h t s e v i g 1 e l b a T

i o t r e d r o n

I llustrate the performance of the proposed algorithm on larger scale problems ,we [

n i d e n i f e d s a e l a c s r e g r a l d n a e l a c s e l d d i m : s p u o r g o w t e d i v o r

p 13] .Two specific problems are

2 e l b a

T displayst heconfiguratio s n oft het estedproblems .A total fo 15000i ndividualsand22500 d

n a e l a c s e l d d i m e h t r o f d e t a r e n e g e r a s l a u d i v i d n

i et h largeronesforeachproblem,r espectively .And m

e l b o r p h c a

e isperformed16runsi ndependently .Wese tp - eop siz be100 forallt ypesofproblems e

l b a T . y l e v i t c e p s e r , s m e l b o r p 3 G d n a 2 G , 1 G r o f s n o i t a r e t i 5 2 2 d n a 5 2 2 , 0 5 1 d n

a 2 givest her esultsof

. e m i t t s r i f e h t r o f s r a e p p a n o i t u l o s t s e b e h t h c i h w t a s n o i t a r e t i e h t f o e u l a v e g a r e v a e h t

e l b a t e h t n i s t l u s e r e h

T 2 showt ha touralgorithmoutperformsGABBontheseexamples.

2 e l b a

T . Theaveragevalueofiterations (EI). :

1

G V=40 EDA GABB

n m q Ind. R un

1 2 8 1 2 1 2 15000 1 6 9 .44 14.33 2 2 0 2 0 2 0 15000 1 6 9 .38 27.27

: 2

G V=60

n m q

3 3 0 3 0 3 0 22500 1 6 14.13 64.53

4 4 2 18 1 8 22500 1 6 14.13 38.03

: 3

G V=100

n m q

5 7 0 3 0 3 0 22500 1 6 18.75 98.70 6 5 0 5 0 5 0 22500 1 6 14.13 86.00

e l b a

T 3. ThemeansofET(s)andTT(s).

: 1

G V=40 MeanE T M_T e_Tan

n m q EDA GABB EDA GABB

1 2 8 1 2 1 2 7.10 3 .01 67.01 28.55

2 2 0 20 2 0 7 .19 10.07 28.05 50.85 l

l

A 7.15 6 .54 47.53 3 9.7

: 2

G V=60

n m q

3 3 0 3 0 3 0 39.38 39.14 111.87 129.71 4 4 2 1 8 1 8 6 .12 10.96 34.88 57.54

l l

A 22.75 25.05 73.38 93.63

: 3

G V=100

n m q

5 7 0 3 0 3 0 8 .98 53.52 99.05 117.18 6 5 0 5 0 5 0 71.92 225.81 459.14 455.87

l l

A 40.45 133.67 279.10 286.53

e l b a

T 3 shows theET and theaverage valueof theCPU time(TT)involved in making al lthe s

n o i t a r e t

i on thetestedproblems .Thelas trow‘All’ineach group reflectstheaveragevalueofthe .

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