) 7 1 0 2 E II A ( g n ir e e n i g n E l a ir t s u d n I d n a e c n e g il l e t n I l a i c if it r A n o e c n e r e f n o C l a n o it a n r e t n I d r 3 7 1 0 2 8 7 9 : N B S
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21Schoo lofMathemaitcsandStaitsitcs ,Qingha iNorma lUniverstiy,Xining810008,China
2Schoo lofComputerScience,Qingha iNorma lUniverstiy,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 Mulit-objecitve opitmizaiton problem, Evoluitonary algortihm, Opitma l soluitons , n o it a m r o f n i m e l b o r P . .t c a r t s b
A Mulit-objectiveoptimizationproblemsareakindofproblemsoptimizingsimutlaneously l a r e v e
s conflicting objectivesand keeping a balancebetween thediversity and theconvergence of s n o i t u l o
s . In this paper , some nove l techniques are designed to improve the efficiency of i
t l u
m -objectiveevolutionaryalgorithms .Firstly,aspecifics -ub funcitonisseparatedfromaseriesof s e v i t c e j b
o ,whichisapplied toprovideanapproximatesearchdirectionand speed theconvergence , n e h T . m h t i r o g l a e h t f
o the crowding degree scheme ,as in NSGA- II ,is used to selec tpotentia l f o s s e c o r p e h t n i s n o i t u l o s g n i s i m o r
p tierationssuch tha tPareto soluiton se thasmoreuniformand n
e t x
e sive distribution .Finally ,a nove lmutli-objective evolutionary algorithm is presented by g n i d d e b m
e these schemes intoMOEA/D .The simulaiton results show the proposed algorithm is b
i s a e
f leandefficien.t
n o it c u d o r t n I i t l u
M -objectiveoptimization si amainresearchi ssuei nopitmizaitonfield[1] .Theyaregenerally n u d n a d e n i a r t s n o c o t n i d e d i v i
d constrained problems .Themathematica lmode lcan beformulated s
a :
( ) ( )
( )
(
1 2)
. , , . , x x x x m t s f f f n i m … Ω
∈ ( 1)
n e h
W Ω=Rn ,(1)i sanunconstrainedproblem ;otherwisei tiscalledaconstrainedoptimization . m e l b o r p u l o s y n a o n s t s i x e e r e h t d n a , s e l b a i r a v n o i s i c e d y b d e d i c e d s i e v i t c e j b o h c a
E tion to
t e s n o i t u l o s e s i m o r p m o c a d n i f o t y r a s s e c e n s ’ t i , e r o f e r e h T . s e v i t c e j b o l l a e z i m i t p o y l s u o e n a t l u m i s it l u m f o n o i t a c i l p p a d a e r p s e d i w d n a y t i x e l p m o c e h t g n i r e d i s n o C . s e v i t c e j b o l l a r o
f -objective
t i r o g l a e v it c e f f e y n a m t n e s e r p s e c n e r e f e r e h t , s m e l b o r
p hmsbasedonvariousoptimizationpurposes .
it l u m e v l o s o t d e s u y l r a e s i m h t i r o g l a t n e i d a r g d e t h g i e w e h
T -objective optimizaiton problems ,for
, e l p m a x
e VEGA[2] ,HLGA[3] ,etc .Thedeficiencyoft hesemethodsi st hatt heycanonlyofferone a m it p o d e r r e f e r
p lsolution whenthealgorithmisexecuted ,and thebes tweigh tvectorisalwaysno t e g r a l h t i w g n il a e d e m i t n o it a t u p m o c e c u d e r o t r e d r o n i n e h T . e c n a v d a n i d e n i m r e t e
d -scale
t n e s e r p e r a s t h g i e w d e x i f g n i s u s m h t i r o g l a n o it i s o p m o c e d e h t , s m e l b o r p n o i t a z i m i t p
o ed , for
, e l p m a x
e MOEA/D[4]andMOEA/DVA[5].
i t l u m g n i v l o s n i y c n e i c i f f e s w o h s n o it i s o p m o c e d n o d e s a b D / A E O
M -objective problems ,bu t
e s u a c e B ) 2 ; w o l s i s n o i t u l o s g n i s i m o r p w e n g n i t t e g f o y c n e i c i f f e e h T ) 1 : s k c a b w a r d e m o s e r a e r e h t h g i e w f o y t i m r o f i n u e h
t tvectors canno tcompletely ensure the uniformity of Pareto front ,the t p i r c s u n a m s i h t , s k c a b w a r d e h t e m o c r e v o o t r e d r o n I . n i a t r e c t o n s i t n o r f o t e r a P f o y t i m r o f i n u s e m e h c s w e n o w t s e s o p o r
p .Firstly ,an sub-function is taken from objectives ,and the descending b u s e h t f o n o it c e r i
d -function is used to generate offspring in evolutionary process .Secondly ,the n o n f o y t i m r o f i n u e h t p e e k o t d e t p o d a y l l a c i d o i r e p s i e m e h c s e e r g e d g n i d w o r
c -dominated soluitons
e m o
S Ba Nisc o itons
n
I m itu -l objectiveoptimizationproblems ,severa lbasicconceptionaredescribedasfollows[1]:
n o it i n if e
D 2.1 (Pareto dominance) F or two ya n solution x1,x2∈Rn i,fthe following conditions
e r
a satisfied:
2 1
2 1
} , , 2 , 1 { , ) ( ) (
} , , 2 , 1 { , ) ( ) (
x x
x x
j j
j j
m j
f f
m j
f f
∈ ∀ ≤
< ∃ ∈
( 2)
n e h
T the vector(f1(x1) ,f2(x1) … , fm(x1) ) is said to dominate thevector (f1(x2) , f2(x2) … , fm(x2) ,)
d n
a x1dominates x2.
n o it i n if e
D 2.2Ift herei snosoluitondominatingx* inRn,t henx*iscalledanoptima lsolution.
n o it i n if e
D 2.3Se t E(f,Rn)={x∈Rn :xisaParetoopitma lsolution}i saParetooptima lse.t
n o it i n if e
D 2.4Se t FE(f,Rn)={f1(x),f2(x),,fm(x)|x∈E(f,Rn)} isaParetofron.t
m h ti r o g l A d e s o p o r P
S bu -funciton Seleciton
a i t n e r e f f i d e l p m i s a t c e l e s e
W b bles -u function fwhich i smonotonoust o some objective function . b
u s e h
T -funciton can beobtained by taking variableoneby one .Themorevariablescan betaken b
u s e h t o t n
i -function ,thebetterthesub-function is .Thegradien tornegativegradien tistaken asa .
n o it a r e p o r e v o s s o r c n i d e s u s i h c i h w n o it c e r i d h c r a e
s Ifthereexis tnodifferentiablesub-funcitons ,
an pa proximategradienti savailablet oserveasapproximatedescendingdireciton.
HybridCrossover
r o
F paren tindividual x0 ,the offspring is denoted by x1 . F or probability threshold s .a random r
e b m u
n r isgenerated .Ifr<s ,then
0
1 x *
x = ±a ∇f (3)
e r e h
W a= -1 i/Gen ,Gen is the maximum number of iterations .a is a monotonic decreasing n
o i t a r e n e g h ti w d e t a i c o s s a n o i t c n u
f i so tha ta big step is needed a tthebeginning of theiteration h
t m o r f e p a c s e d n a s l a u d i v i d n i r e t t e b d n i f o t r e d r o n i s s e c o r
p e loca loptima ;A tthe end of the
n e e w t e b n o i t a l e r e h t f I . n o it u l o s l a m i t p o e s i c e r p e h t n i a t b o o t d e t p o d a s i h c r a e s g n i n i f e r , m h t i r o g l a
b u s d n a n o i t c n u f l a n i g i r
o -function is monotonic increasing , x1= x0−a*∇f ; otherwise ,
0
1 x *
x = +a ∇f .When r≥s,t hecrossoveroperatori st akenast hesamei n[4].
CrowdingDegree
e c n i
S the uniformtiy of weigh tvectors of MOEA/D canno tensure the uniformity of the Pareto t
e s n o i t u l o
s ,andthecomputationamoun tislargei hft ecrowdingdegreemethodisusedin each of s
a s n o i t a r e n e
g i n NSGA- II . Therefore, in the proposed algorithm ,we adop t periodically the e
e r g e d g n i d w o r
c methodofNSGA-IIt okeeptheuniformityoft heParetosolutionse.t
M itu -l objecitveEvolu itonaryAlgorit hmBasedSpeci ifcProblem
e v o b a e h t g n i d d e b m
E schemes into MOEA/D ,a problem-specific mulit-objective evolutionary ,
m h t i r o g l
a P-MOEA/D, isdescribedasfollows.
p e t
S 1 .Generatingani niita lpopulation . t e s l a u d i v i d n i e t a r e n e g y l m o d n a
R {x1,x2,,xPopsize}as an initia lpopulation ,and denote the n
o
n -dominatedsolutionse tbynonD.
p e t
S 2 .Iniitailzingweigh tvectors .
t s e r a e n e h t d n i f d n a , n o i t u b i r t s i d m r o f i n u h t i w s r o t c e v t h g i e w f o p u o r g a e t a e r
C Tweightst oeach
i l c u E o t g n i d r o c c a r o t c e v t h g i e
p e t
S 3 .Evolutionpopulaiton. b
u s a t c a r t x
E -function f according to theabove designs ,generate crossoveroffspring by hybrid s
u a G g n i t u c e x e y b g n i r p s f f o n o i t a t u m n i a t b o d n a r o t a r e p o r e v o s s o r
c smutation.
p e t
S 4 .Selection. n o i t a r e n e g r o
F iandani ntegermgiven,i f imodm==0,t hecrowdingdegreei sexecutedt oselec t n
o
n -dominated solutions ,otherwise ,the nex tgeneration population is chosen in terms of the c
n u f e v i t c e j b o e h t f o m u s d e t h g i e
w tion .Updatethenon-dominatedsolutionse tnonD.
p e t
S 5 .Terminaitoncriterion.
t u p t u o , p o t s , t e m s i n o i r e t i r c n o i t a n i m r e t e h t f
I nonD ;otherwise ,returnt oStep3.
P-MOEA/DdistinguishesfromMOEA/Dinatl eastt woaspects: )
1 Hybrid crossoveroperatorbasedon thedescending direction ofthesub-function isproposed , l
u f p l e h s i h c i h
w togeneratepromisingsoluitons. )
2 The crowding technology is only used periodically instead of being executed in each of ,
s n o i t a r e t
i whichcansavecomputaitonalt ime.
m u
N era lSimula itonandAnalyssi
f o t n e m n o r i v n e n a g n i v a h C P n a n o d e m r o f r e p e r a s t n e m i r e p x e e h
T i -5 2450MCPUwtih2.50GHz.
f o e l a c s e h t : ] 8 [ n i s a s r e t e m a r a p e m a s e h t e k a t e w , D / A E O M h ti w n o s i r a p m o c f o e s o p r u p e h t r o F
n o i t a l u p o
p Popszie=101,scalingfactorF=0.5,t hedimensionofvariablesi nZDT1
�ZDT2
�ZDT3i s
n=30 , whlie n=10 in ZDT4 and ZDT6 ; mutation probabliity prob=1/n ,crossover probabiltiy
R
C =0.5 ,thenumberofthecloses tweightsto each weigh tvectorT=20 ,thenumberofindependen t l
a e h t f o n o i t u c e x
e gorithm numrun=20 .The generation in MOEA/D is Gen=200 ,while Gen=150 P
r o
f -MOEA/D.
: ] 7 [ n e v i g e r a s e l p m a x e e v i F . s e v it c e j b o e h t f o n o i t a z i m i n i m e r a s e c n a t s n i t s e t l l A
: 1 T D
Z f1(x)=x1 , 2 1
) (
] 1
[ * ) ( ) (
) (
x x
x
x f g
f
g −
= ,
e r e h
w 2
) ( * 9 1 ) (
1
x
n
i i
x g
n= + =
−
∑
, x=(x1,x2,,xn)T∈[0,1]n,n=30.
: 2 T D
Z f1(x)=x1 , 1 2
2
)
( ) ]
( 1 [ * ) ( ) (
) (
x x
x
x f g
f
g −
= ,
( g e r e
h x) i sthesameasi nZDT1.
: 3 T D
Z f1(x)=x1 , 1 1
1 2
) ( ) (
] ) * * 0 1 ( n i s * 1
[ * ) ( ) (
) ( ) (
x x x
x
x x
f
f x
g f
g
g − π
−
= ,
( g e r e h
w x) i sthesameasi nZDT1
: 4 T D
Z f1(x)=x1 , 1
2
) (
] 1
[ * ) ( ) (
) (
x x
x
x f g
f
g −
= ,
e r e h
w 2
2
] ) * * 4 ( s o c * 0 1 [ ) 1 ( * 0 1 1 )
(x n i i
i
x x
n
g π
=
− +
− +
=
∑
12
1, , , ) [0,1] [ 5,5] , 10
(
x T n
n n
x x
x ∈ × − − =
= .
: 6 T D
Z f1(x)=1−exp(−4*x1)*sin(6*π*x1), 2 1 2 ) (
] ) ( 1 [ * ) ( ) (
) (
x x
x
x f g
f
g −
= ,
e r e h
w 2 0.25
) (
] [
* 9 1 ) (
1
x
n i i
x g
n= + =
−
∑
,x ( 1, 2, , )T [0,1]n, 10
n n
x x
x ∈ =
= .
C-metric and Hyper-volume (HV) are selected to compare the n -on dominated sets .We firslty f
o e u l a v e h t e n i m r e t e
d s by an experimen tmethod onZDT1 and ZDT2 ;secondly ,makeuseofthe d
e n i m r e t e
. D / A E O M h t i w V
H
) i
( C- metric[8]
D / A E O M f o t e s n o i t u l o s l a m it p o o t e r a P : A t e
S ;
P f o t e s n o i t u l o s l a m it p o o t e r a P : B t e
S -MOEA/D
| } :
| { | ) , (
| |
v v
u B A dominatesu B
A C
B ∈ ∃ ∈
= ( 4)
C (A ,B) is defined as the percentage of the solutions in B tha tare dominated by a tleas tone n
i n o i t u l o
s A. C(A ,B)i sno tnecessarilyequalt o1-C(B ,A). C(A ,B)=1meanst ha tal lsolutionsi n
B are dominated by some solutions in A ,whlie C (A ,B)=0 implies tha tno solution in B is o
i t u l o s a y b d e t a n i m o
d ni nA.
) i i
( HV[8]
e c a p s e v it c e j b o e h t n i t n i o p e c n e r e f e r a t c e l e
S r=(r1,r2,…,rm)T ,compute
V
H ( ,r) ( [ 1, 1] [ m, m])
A f
r f r
f e m l o v A
∈
× ×
=
( 5)t n i o p e c n e r e f e r a r o
F r,t hel agerHVvaluemeansbetterqualtiy . )
i i i
( Selectedsub-function
Table1 .Sub-functionsusedi nP-MOEA/D. .
b o r
P S -ubfunctionf Relations 1
T D Z
2 x
n i i=
∑
f2 :monotonei ncreasing2 T D Z
2 x
n i i=
∑
f2 :monotonei ncreasing3 T D Z
2 x
n i i=
∑
f2 :monotonei ncreasing4 T D
Z 2
2
] ) * * 4 ( s o c * 0 1 [ n
i i
i
x
x π
=
−
∑
f2 :monotonei ncreasing6 T D Z
2
x
n i i=
∑ f2 :monotonei ncreasing
We take s=0 .1, 0 .3, 0 .5, and 0.7 f orZDT1-2 for thesakeof selecting appropriates .Resutlsare :
2 e l b a T s a d e b i r c s e d
. 2 e l b a
T Resultsabou tdifferents no ZDT1-2.
s C(A, B) C(B, A) C(A, B) C (B, A) T
D
Z 1 ZDT2
1 .
0 0.9802 0.9901 0.9901 0.9802 3
.
0 0.9802 0.9901 0.9802 0.9901 5
.
0 0.9802 0.9901 0.9802 0.9901 7
.
0 0.9802 0.9901 0.9901 0.9802
2 e l b a
T clearly indicatestha tforZDT1 ,whens=0.1,0.3,0.5,0.7 ,C (A, B) <C (B ,A).ForZDT2, n
e h
w s=0.3and0.5, C(A, B) <C(B, A) .Therefore,wetakes=0.5 . n
e h
W s=0.5 ,wecarryou tP-MOEA/D tota l20runsand providetheaveragevaluesofC-metric V
H d n
a inTable3.
e l b a
T 3. ResultscomparisonwithMOEA/D.
r
Po b. MOEA/DG enP-MOEA/D MOEA/DCPUP-MOEA/D MOEA/D H VP-MOEA/D C(A,B) C(B,A) 1
T D
Z 2 00 1 50 10.0870 7.6222 0.0944 0.0945 0.9802 0.9901 2
T D
Z 2 00 1 50 10.0059 7.6745 9.6847e- 50 9.7060e- 50 0.9802 0.9901 3
T D
Z 2 00 1 50 9.8444 7.7245 0.1136 0.1113 0.9802 0.980 2 4
T D
Z 2 00 1 50 9.8561 7.2501 0.0819 0.0900 0.9802 0.9802 6
T D
Table3 showsaverageindexvalueof20runsaswel lasgenerationnumbers .HVofP-MOEA/D D
/ A E O M n a h t r e g r a l s
i on al lproblems excep tfor ZDT3 .C (A, B) < C (B, A) holds on al ltes t s
m e l b o r
p ,i tmeanst ha tP-MOEA/Dfoundt hebetternon-dominatedsett hanMOEA/D. I tshouldbe P
t a h t d e t o
n -MOEAexecutedal essnumberofgenerationst hanMOEA/D.
Conclu ison
k a t y
B i ng a sub-funciton from original objecitve functions w , e propose a new multi-objecitve m
h t i r o g l a y r a n o i t u l o v
e on the basis of the existing MOEA/D .Since the objective information is d
e z i l i t
u sucht hatt heproposedalgorithmcanfindpromisingi ndividualsi niteraitons .Inaddition ,in r
o der to keep theuniform Pareto front ,the crowding degree scheme is also adopted periodically . i
c i f f e e h t e t a r t s u l li s t n e m i r e p x e e h t f o s t l u s e r e h
T encyoft heproposedalgorithm.
Acknowledgements
e i c S l a r u t a N l a n o it a N e h t y b d e t r o p p u s s a w k r o w h c r a e s e r e h
T n ceFoundaitonofChinaunderGran t
1 6 . o
N 463045 and the Key Laboratory of the Interne t of Things of Qingha i e
c n i v o r
P (2017- JZ -Y21) .
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