Computing the Set of Approximate Solutions of an MOP with Stochastic Search Algorithms

(1)

MOP with Stochastic Search Algorithms

Oliver Schuetze, Carlos A. Coello Coello, El-Ghazali Talbi

To cite this version:

Oliver Schuetze, Carlos A. Coello Coello, El-Ghazali Talbi. Computing the Set of Approximate

Solutions of an MOP with Stochastic Search Algorithms.

research report.

2007.

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inria-00157741, version 2 - 23 Oct 2007

a p p o r t

d e r e c h e r c h e

9 -6

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9

9 IS

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IN

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--?

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--F

R

+

E

N

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Thèmes COM et COG et SYM et NUM et BIO

Computing the Set of Approximate Solutions of an

MOP with Stochastic Search Algorithms

Oliver Schütze

1 , Carlos A. Coello Coello

2 , and El-Ghazali Talbi

1

1 _{INRIA Futurs, LIFL, CNRS Bât M3, Cité Scientifique}

e-mail: {schuetze,talbi}@lifl.fr

3 _{CINVESTAV-IPN, Electrical Engineering Department}

e-mail: ccoello@cs.cinvestav.mx

N° ????

(3)

(4)

Unité de recherche INRIA Futurs

OliverS hütze

1

, CarlosA. CoelloCoello

2

, and El-GhazaliTalbi

1

INRIAFuturs, LIFL, CNRS Bât M3,Cité S ientique

e-mail: {s huetze,talbi}li.fr

3

CINVESTAV-IPN, Ele tri alEngineering Department

e-mail: oello s. investav.mx

ThèmesCOMetCOGet SYMetNUM etBIOSystèmes ommuni antsetSystèmes

ognitifsetSystèmessymboliquesetSystèmesnumériquesetSystèmesbiologiques

ProjetsApi set Opéra

Rapportdere her he n°???? June200720pages

Abstra t: Inthisworkwedevelopaframeworkfortheapproximationof theentire setof

ǫ

-e ient solutionsof a multi-obje tiveoptimization problem with sto hasti sear h

algo-rithms. For this,weproposethesetofinterest,investigateitstopologyandstatea

onver-gen eresultforageneri sto hasti sear halgorithmtowardthissetofinterest. Finally,we

presentsomenumeri alresultsindi atingthepra ti abilityofthenovelapproa h.

Key-words: multi-obje tiveoptimization, onvergen e,

ǫ

-e ientsolutions,approximate

solutions,sto hasti sear halgorithms.

∗

Partsofthismanus riptwillbepublishedinthepro eedingsofthe6thMexi anInternationalConferen e

(5)

Résumé: Pasderésumé

(6)

1 Introdu tion

Sin ethenotionof

ǫ

-e ien yformulti-obje tiveoptimizationproblems(MOPs)hasbeen

introdu edmorethantwode adesago([7℄),this on epthasbeenstudiedandusedbymany

resear hers,e.g. to allow (or tolerate)nearlyoptimal solutions([7℄, [16℄), to approximate

thesetofoptimalsolutions([10℄),orinordertodis retizethisset([6℄,[14℄).

ǫ

-e ient

solu-tionsorapproximatesolutionshavealsobeenusedtota kleavarietyofrealworldproblems

in ludingportfoliosele tionproblems([17℄),alo ationproblem([1℄),oraminimal ostow

problem([10℄). Theexpli it omputationofsu happroximatesolutionshasbeenaddressed

in severalstudies (e.g.,[16℄, [1℄, [3℄),in allof thems alarizationte hniqueshavebeen

em-ployed.

The s opeof this paper is to develop aframework for the approximationof the set of

ǫ

-e ient solutions (denote by

E

ǫ

) with sto hasti sear h algorithms su h as evolutionary

multi-obje tive(EMO)algorithms. This allsforthedesignofanovelar hivingstrategyto

storethe`required'solutionsfoundbyasto hasti sear hpro ess(thoughtheinvestigation

ofthe set ofinterestwill bethemajorpartin this work). Oneinterestingfa t isthat the

solutionset (the Pareto set) is in ludedin

E

ǫ

for all (small)values of

ǫ

, and thus the

re-sultingar hivingstrategyforEMOalgorithms anberegardedasanalternativetoexisting

methodsfortheapproximationofthisset (e.g,[4℄,[8℄, [5℄,[9℄).

Theremainder ofthis paperisorganizedasfollows: in Se tion 2,wegivetherequired

ba kgroundfortheunderstandingofthesequel. InSe tion 3,weproposeaset ofinterest,

analyzeits topology, andstate a onvergen eresult. Wepresentnumeri alresultson two

examplesin Se tion4and on ludeinSe tion 5.

2 Ba kground

Inthefollowingwe onsider ontinuousmulti-obje tiveoptimizationproblems

min

x∈

R

n

{F (x)},

(MOP)

where the fun tion

F

is dened as the ve tor of the obje tive fun tions

F :

R

n

_→

R

k

_,

_{F (x) = (f}

1 (x), . . . , f

k

(x)),

and where ea h

f

i

:

R

n

_→

R is ontinuously

dieren-tiable. Laterwewill restri tthesear htoa ompa tset

Q ⊂

R

n

,thereadermaythink of

an

n

-dimensionalbox.

Denition2.1 (a) Let

v, w ∈

R

k

. Then the ve tor

v

islessthan

w

(

v <

p

w

), if

v

i

< w

i

for all

i ∈ {1, . . . , k}

. The relation

≤

p

isdenedanalogously.

(b)

y ∈

R

n

isdominatedbyapoint

x ∈

R

n

(

x ≺ y

)withrespe tto(MOP)if

F (x) ≤

p

F (y)

and

F (x) 6= F (y)

,else

y

is alled nondominatedby

x

.

( )

x ∈

R

n

is alledaParetopointifthereisno

y ∈

R

n

(7)

(d)

x ∈

R

n

is weakly Pareto optimal if there does not exist another point

y ∈

R

n

su h that

F (y) <

p

F (x)

.

Wenowdeneaweaker on eptofdominan e, alled

ǫ

-dominan e,whi histhebasisof

theapproximation on eptusedin thisstudy.

Denition2.2 Let

ǫ = (ǫ

1 , . . . , ǫ

k

) ∈

R

k

+

and

x, y ∈

R

n

.

x

issaidto

ǫ

-dominate

y

(

x ≺

ǫ

y

)

withrespe tto(MOP )if

F (x) − ǫ ≤

p

F (y)

and

F (x) − ǫ 6= F (y)

.

Theorem2.3([11℄) Let(MOP)begivenand

q :

R

n

_→

R

n

bedenedby

q(x) =

P

k

i=1

α

ˆ

i

∇f

i

(x)

, where

α

ˆ

isasolutionof

min

α∈

R

k

(

k

X

i=1

α

i

∇f

i

(x)k

2

2 ; α

i

≥ 0, i = 1, . . . , k,

k

X

i=1

α

i

= 1

)

.

Then either

q(x) = 0

or

−q(x)

is a des ent dire tion for all obje tive fun tions

f

1 , . . . , f

k

in

x

. Hen e, ea h

x

with

q(x) = 0

fullls the rst-order ne essary ondition for Pareto

optimality.

In ase

q(x) 6= 0

itobviouslyfollowsthat

q(x)

isanas entdire tionforallobje tives. Next,

weneedthefollowingdistan es betweendierentsets.

Denition2.4 Let

u ∈

R

n

and

A, B ⊂

R

n

. The semi-distan e dist

(·, ·)

andthe Hausdor

distan e

d

H

(·, ·)

aredenedasfollows:

(a) dist

(u, A) := inf

v∈A

ku − vk

(b) dist

(B, A) := sup

u∈B

dist

(u, A)

( )

d

H

(A, B) := max {

dist

(A, B),

dist

(B, A)}

Denoteby

A

the losureofaset

A ∈

R

n

,by

◦

A

itsinterior,andby

∂A = A\

◦

A

theboundary

of

A

.

Algorithm 1givesaframework of ageneri sto hasti multi-obje tiveoptimization

al-gorithm,whi h will be onsidered in this work. Here,

Q ⊂

R

n

denotesthe domain ofthe

MOP,

P

j

the andidateset(orpopulation)ofthegenerationpro essatiterationstep

j

,and

A

j

the orrespondingar hive.

Denition2.5 A set

S ⊂

R

n

is alled not onne tedif there existopen sets

O

1 , O

2

su h

that

S ⊂ O

1 ∪ O

2

,

S ∩ O

1 6= ∅

,

S ∩ O

2 6= ∅

,and

S ∩ O

1 ∩ O

2 = ∅

. Otherwise,

S

is alled onne ted.

(8)

Algorithm1Generi Sto hasti Sear hAlgorithm 1:

P

0 ⊂ Q

drawnatrandom 2:

A

0 = ArchiveU pdate(P

0 , ∅)

3: for

j = 0, 1, 2, . . .

do 4:

P

j+1

= Generate(P

j

)

5:

A

j+1

= ArchiveU pdate(P

j+1

, A

j

)

6: endfor

3 The Ar hiving Strategy

Inthisse tionwedenethesetofinterest,investigatethetopologyofthisobje t,andnally

statea onvergen eresult.

Denition3.1 Let

ǫ ∈

R

k

+

and

x, y ∈

R

n

.

x

is said to

−ǫ

-dominate

y

(

x ≺

−ǫ

y

) with

respe tto(MOP) if

F (x) + ǫ ≤

p

F (y)

and

F (x) + ǫ 6= F (y)

.

Thisdenitionisof ourseanalogoustothe` lassi al'

ǫ

-dominan erelationbutwithavalue

˜

ǫ ∈

R

k

−

. However,wehighlightit heresin eit willbeused frequently in thiswork. While

the

ǫ

-dominan eisaweaker on eptofdominan e,

−ǫ

-dominan eisastrongerone.

Denition3.2 Apoint

x ∈ Q

is alled

−ǫ

weakParetopointifthereexistsnopoint

y ∈ Q

su hthat

F (y) + ǫ <

p

F (x)

.

Nowweareabletodenethesetofinterest. Ideally,wewouldliketoobtainthe` lassi al'

set

P

c

Q,ǫ

:= {x ∈ Q|∃p ∈ P

Q

: x ≺

ǫ

p}

1

,

(1)

where

P

Q

denotestheParetoset(i.e.,thesetofParetooptimalsolutions)of

F

Q

. Thatis,

everypoint

x ∈ P

c

Q,ǫ

is` lose'toatleastonee ientsolution,measuredinobje tivespa e.

However,sin ethissetisnoteasyto at hnotethatthee ientsolutionsareusedinthe

denition,wewill onsideranenlargedsetofinterest(seeLemma3.9):

Denition3.3 Denote by

P

Q,ǫ

the set of points in

Q ⊂

R

n

whi h are not

−ǫ

-dominated

byany other pointin

Q

,i.e.

P

Q,ǫ

:= {x ∈ Q| 6 ∃y ∈ Q : y ≺

−ǫ

x}

2

(2)

Example3.4 (a) Figure1shows twoexamples forsets

P

Q,ǫ

,onefor the single-obje tive

ase(left), andonefor

k = 2

(right). Inthe rst asewehave

P

Q,ǫ

= [a, b] ∪ [c, d]

.

1

P

c

Q,ǫ

is loselyrelatedtoset

E

1

onsideredin[16℄.

2

P

Q,ǫ

is loselyrelatedtoset

E

5

(9)

Figure1: Twodierentexamplesforsets

P

Q,ǫ

. Left for

k = 1

andinparameterspa e,and

rightanexamplefor

k = 2

inimagespa e.

(b) Consider the MOP

F :

R

→

R

2 _,

_{F (x) = ((x − 1)}

2 _{, (x + 1)}

2 ₎

. For

ǫ = (1, 1)

and

Q

su iently large, say

Q = [−3, 3]

, we obtain

P

Q

= [−1, 1]

and

P

Q,ǫ

= (−2, 2)

. Note

that the boundary of

P

Q,ǫ

,i.e.

∂P

Q,ǫ

= P

Q,ǫ

\

◦

P

Q,ǫ

= [−2, 2]\(2, 2) = {−2, 2},

isgiven

by

−ǫ

weak Pareto points whi h are not in luded in

P

Q,ǫ

(see also Lemma 1): for

x

1 = −2

and

x

2 = 2

it is

F (x

1 ) = (9, 1)

and

F (x

2 ) = (1, 9)

. Sin e there exists no

x ∈ Q

with

f

i

(x) < 0, i = 1, 2

,thereisalsonopoint

x ∈ Q

whereallobje tivesareless

than at

x

1

or

x

2

. Further,sin e

F (−1) = (4, 0)

and

F (1) = (0, 4)

there exist points

whi h

−ǫ

-dominatethese points, andtheyarethusnot in ludedin

P

Q,ǫ

.

Firstwestudythe onne tednessof

P

Q,ǫ

. The onne tednessoftheset ofinterestisan

importantproperty,inparti ularwhenta klingtheproblemwithlo alsear hstrategies: in

that ase,theentireset anpossiblybedete tedwhenstartingwithonesingleapproximate

solution. Example3.4(a)showsthatthe onne tednessof

P

Q,ǫ

annotbeexpe tedin ase

F (Q)

isnot onvex. However,inthe onvex asethefollowingholds.

Lemma3.5 Let

ǫ ∈

R

k

+

. If

Q

is onvexand all

f

i

, i = 1, . . . , k

are onvex,then

P

Q,ǫ

and

F (P

Q,ǫ

)

are onne ted.

Proof: The proof is based on the fa t that in this ase

P

Q

is onne ted (e.g., [2℄).

Assume that

P

Q,ǫ

is not onne ted, that is, there exist open sets

O

1 , O

2 ⊂

R

n

su h that

P

Q,ǫ

⊂ O

1 ∪O

2

,

P

Q,ǫ

∩O

1 6= ∅

,

P

Q,ǫ

∩O

2 6= ∅

,and

P

Q,ǫ

∩O

1 ∩O

2 = ∅

. Sin e

P

Q

is onne ted,

itmustbe ontainedinoneofthesesets,withoutlossofgeneralityweassumethat

P

Q

⊂ O

1

.

Byassumptionthereexistsapoint

x ∈ P

Q,ǫ

∩ O

2

,andhen e,

x 6∈ P

Q

. Further,thereexists

anelement

p ∈ P

Q

su hthat

F (p) ≤

p

F (x)

. Sin e

Q

is onvex,thefollowingpath

γ : [0, 1] → Q

(10)

PSfragrepla ements

x

p

O

1 O

2 γ([0, 1])

P

Q,ǫ

Figure2: ...

iswell-dened (i.e.,

γ([0, 1]) ⊂ Q

). Sin e

F

is onvexand bythe hoi eof

p

itholdsforall

λ ∈ [0, 1]

:

F (γ(λ)) = F (λx + (1 − λ)p) ≤ λF (x) + (1 − λ)F (p) ≤ F (x),

(4) andthusthat

γ(λ) ∈ P

Q,ǫ

,whi hisa ontradi tiontothe hoi eof

O

1

and

O

2

.

The onne tedness of

F (P

Q,ǫ

)

follows immediately sin e images of onne ted sets under

ontinuousmappingsare onne ted,andtheproofis omplete.

Thenextlemmades ribesthetopologyof

P

Q,ǫ

inthegeneral ase,whi hwillbeneeded

fortheup oming onvergen eanalysis ofthesto hasti sear hpro ess.

Lemma3.6 (a) Let

Q ⊂

R

n

be ompa t. Under the following assumptions

(A1) Letthere beno weak Pareto pointin

Q\P

Q

,where

P

Q

denotes the setof Pareto

pointsof

F |

Q

.

(A2) Lettherebeno

−ǫ

weak Pareto pointin

Q\P

Q,ǫ

,

(A3) Dene

B := {x ∈ Q|∃y ∈ P

Q

: F (y) + ǫ = F (x)}

. Let

B ⊂

◦

Q

and

q(x) 6= 0

for all

x ∈ B

,where

q

isasdenedin Theorem 2.3 ,

itholds:

P

Q,ǫ

= {x ∈ Q| 6 ∃y ∈ Q : F (y) + ǫ <

p

F (x)}

◦

P

Q,ǫ

= {x ∈ Q| 6 ∃y ∈ Q : F (y) + ǫ ≤

p

F (x)}

∂P

Q,ǫ

= {x ∈ Q|∃y

1 ∈ P

Q

: F (y

1 ) + ǫ ≤

p

F (x) ∧ 6 ∃y

2 ∈ Q : F (y

2 ) + ǫ <

p

F (x)}

(5)

(b) Letinaddition tothe assumptionsmade abovebe

q(x) 6= 0 ∀x ∈ ∂P

Q,ǫ

. Then

◦

(11)

Proof: Dene

W := {x ∈ Q| 6 ∃y ∈ Q : F (y) + ǫ <

p

F (x)}

. We show the equality

P

Q,ǫ

= W

by mutual in lusion.

W ⊂ P

Q,ǫ

follows dire tly by assumption (A2). To see

theother in lusion assumethat there existsan

x ∈ P

Q,ǫ

\W

. Sin e

x 6∈ W

there existsan

y ∈ Q

su h that

F (y) + ǫ <

p

F (x)

. Further, sin e

F

is ontinuous there exists further a

neighborhood

U

of

x

su hthat

F (y) + ǫ <

p

F (u),

∀u ∈ U

. Thus,

y

is

−ǫ

-dominatingall

u ∈ U

(i.e.,

U ∩ P

Q,ǫ

= ∅

),a ontradi tiontotheassumptionthat

x ∈ P

Q,ǫ

. Thus, wehave

P

Q,ǫ

= W

as laimed.

Nextweshowthat theinteriorof

P

Q,ǫ

isgivenby

I := {x ∈ Q| 6 ∃y ∈ Q : F (y) + ǫ ≤

p

F (x)},

(7)

whi h we doagainby mutualin lusion. Toseethat

◦

P

Q,ǫ

⊂ I

assumethat there existsan

x ∈

◦

P

Q,ǫ

\I

. Sin e

x 6∈ I

wehave

∃y

1 ∈ Q :

F (y

1 ) + ǫ ≤

p

F (x).

(8) Sin e

x ∈

◦

P

Q,ǫ

there exists no

y ∈ Q

whi h

−ǫ

-dominates

x

, and hen e,equality holds in

equation(8). Further,byassumption(A1) itfollowsthat

y

1

mustbein

P

Q

. Thus,we an

reformulate(8)by

∃y

1 ∈ P

Q

:

F (y

1 ) + ǫ = F (x)

(9)

Sin e

x ∈

◦

P

Q,ǫ

there exists a neighborhood

U

˜

of

x

su h that

U ⊂

˜

◦

P

Q,ǫ

. Further, sin e

q(x) 6= 0

byassumption(A1),thereexistsapoint

x ∈ ˜

˜

U

su hthat

F (˜

x) >

p

F (x)

. Combining

thisand(9)weobtain

F (y

1 ) + ǫ = F (x) <

p

F (˜

x),

(10)

andthus

y

1 ≺

−ǫ

x ∈ ˜

˜

U ⊂

◦

P

Q,ǫ

, whi his a ontradi tion. Itremainstoshowthat

I ⊂

◦

P

Q,ǫ

:

assumethereexistsan

x ∈ I\

◦

P

Q,ǫ

. Sin e

x 6∈

◦

P

Q,ǫ

thereexistsasequen e

x

i

∈ Q\P

Q,ǫ

, i ∈

N

,

su h that

lim

i→∞

x

i

= x

. That is, there exists asequen e

y

i

∈ Q

su hthat

y

i

≺

−ǫ

x

i

for

all

i ∈

N. Sin e all the

y

i

are inside

Q

, whi h is a bounded set, there exists a

subse-quen e

y

i

j

, j ∈

N, and an

y ∈ Q

su h that

lim

j→∞

y

i

j

= y

(Bolzano-Weierstrass). Sin e

F (y

i

j

) + ǫ ≤

p

F (x

i

j

),

∀j ∈

N

,

itfollowsfor thelimitpoints thatalso

F (y) + ǫ ≤

p

F (x)

,

whi h isa ontradi tionto

x ∈ I

. Thus, wehave

◦

P

Q,ǫ

= Iasdesired.

Fortheboundaryweobtain

∂P

Q,ǫ

= P

Q,ǫ

\

◦

P

Q,ǫ

= {x ∈ Q|∃y

1 ∈ Q : F (y

1 ) + ǫ ≤

p

F (x)

and

6 ∃y

2 ∈ Q : F (y

2 ) + ǫ <

p

F (x)}

(11)

Sin eby(A1)thepoint

y

1

in(11)mustbein

P

Q

,weobtain

(12)

Itremainsto showthese ond laim. Itis

P

Q,ǫ

=

◦

P

Q,ǫ

∪ ∂P

Q,ǫ

. Assume that

◦

P

Q,ǫ

6= P

Q,ǫ

,

i.e.,thatthereexistsan

x ∈ ∂P

Q,ǫ

andaneighborhood

U

of

x

su hthat

U ∩

◦

P

Q,ǫ

= ∅

. Sin e

x ∈ ∂P

Q,ǫ

there exists apoint

y ∈ P

Q

su h that

F (y) + ǫ ≤

p

F (x)

. Byassumption it is

q(x) 6= 0

, and thus there exists an

x ∈ U

¯

su h that

F (¯

x) <

p

F (x)

. Sin e

x 6∈

¯

◦

P

Q,ǫ

there

existsan

¯

y ∈ Q

su hthat

F (¯

y) + ǫ ≤

p

F (¯

x) <

p

F (x)

,whi h ontradi tstheassumptionthat

x ∈ ∂P

Q,ǫ

. Thus, wehavethat the losureof theinteriorof

P

Q,ǫ

isequalto its losure as laimed.

Remark3.7 (a) Notethat ingeneral

P

Q,ǫ

isneitheran open nora losed set,andthat

P

Q,ǫ

gets` ompleted'by

−ǫ

weak Pareto points(seealso Example 1).

(b) Sin efor

x

and

y

1

inequation(12)itmustholdthatthereexistsanindex

j ∈ {1, . . . , k}

su hthat

f

j

(y

1 ) + ǫ

j

= f

j

(x)

. Thus, theboundaryof

P

Q,ǫ

an be hara terizedby the

set of

−ǫ

weak Pareto pointswhi h areboundedin obje tivespa efrom

P

Q

by

ǫ

.

The next example shows that the losure of the interior of

P

Q,ǫ

does in general not

havetobeequalto its losure,whi h ausestroubleto approximate

∂P

Q,ǫ

usingsto hasti

sear halgorithms. However,thefollowingLemmashowsthatthisisdespitefortheoreti al

investigationsnotproblemati sin e

◦

P

Q,ǫ

,whi h anbeapproximatedinany ase,already

ontainsalltheinterestingparts.

Example3.8 Figure3shows an examplewhi h is amodi ation of the MOPin Example

3.4(a). Wehave

P

Q,ǫ

= {x

∗

_{} ∪ [c, d]}

andhen e

◦

P

Q,ǫ

= [c, d] 6= P

Q,ǫ

.

Note that here we have

f

′

_(x

∗

_{) = 0}

, and thus that (A3) is violated. The problem with the

approximation of the entire set

P

Q,ǫ

in this ase is the following: assume that argmin

f

is

alreadyamemberofthear hive,thenevery andidatesolutionnear

x

∗

willbereje tedbyall

furtherar hives. Thus, the entireset

P

Q,ǫ

anonly beapproximatedif

x

∗

isamember ofa

population

P

i

, i ∈

N, andthe probability for this eventiszero. Su hproblems do not o ur

for points in

◦

P

Q,ǫ

(seeproof ofTheorem3.10).

Lemma3.9

P

c

Q,ǫ

⊂

◦

P

Q,ǫ

Proof: Assume there exists an

x ∈ P

c

Q,ǫ

\

◦

P

Q,ǫ

. Sin e

x ∈ P

c

Q,ǫ

there exists a Pareto

optimalpoint

p ∈ P

Q

with

p ≺

ǫ

p

. Further,sin e

x 6∈

◦

P

Q,ǫ

there existsan

y ∈ Q

su h that

F (y) + ǫ ≤

p

F (x)

. Combiningbothweobtain

F (y) ≤

p

F (x) − ǫ ≤ F (p),

and

∃j ∈ {1, . . . , k} : f

j

(y) ≤ f

j

(x) − ǫ < f

j

(p) (⇒ F (y) 6= F (p)),

(13)

Figure3: Exampleofaset

P

Q,ǫ

wherethe losureofitsinteriorisnotequalto its losure.

whi h meansthat

y ≺ p

,a ontradi tionto

p ∈ P

Q

,andwearedone.

Havinganalyzedthetopologyof

P

Q,ǫ

weare nowin thepositionto statethefollowing

result. Thear hiving strategyissimplythe onewhi hkeepsallobtainedpointswhi h are

not

−ǫ

-dominatedbyanyothertestpoint.

Theorem3.10 LetanMOP

F :

R

n

_→

R

k

begiven,where

F

is ontinuous,let

Q ⊂

R

n

be

a ompa tset and

ǫ ∈

R

k

+

. Furtherlet

∀x ∈ Q

and

∀δ > 0 :

P (∃l ∈

N

: P

l

∩ B

δ

(x) ∩ Q 6= ∅) = 1

(14)

Then,under the assumptions madeinLemma3.6 , an appli ationof Algorithm1 , where

ArchiveU pdateP

Q,ǫ

(P, A) := {x ∈ P ∪ A : y 6≺

−ǫ

x ∀y ∈ P ∪ A},

(15)

isusedtoupdatethe ar hive, leads toasequen eof ar hives

A

l

, l ∈

N, with

lim

l→∞

d

H

(P

Q,ǫ

, A

l

) = 0,

withprobability one. (16)

Proof: Sin e

dist(A, B) = dist(A, B)

for allsets

A, B ⊂

R

n

and sin e

◦

P

Q,ǫ

= P

Q,ǫ

(see

Lemma3.6),itissu ienttoshowthattheHausdordistan ebetween

A

l

and

◦

P

Q,ǫ

vanishes

inthelimitwithprobabilityone.

Firstweshowthat

dist(A

l

,

◦

P

Q,ǫ

) → 0

withprobabilityonefor

l → ∞

. Itis

dist(A

l

,

◦

P

Q,ǫ

) = max

a∈A

l

inf

p∈

P

Q,ǫ

◦

ka − pk.

Wehavetoshowthatevery

x ∈ Q\P

Q,ǫ

willbedis arded(ifaddedbefore)fromthear hive

afternitelymanysteps,andthatthispointwillneverbeenaddedfurtheron.

(14)

point

p ∈ P

Q,ǫ

su h that

F (p) + ǫ <

p

F (x)

. Further, sin e

F

is ontinuous there exists a

neighborhood

U

of

x

su hthat

F (p) + ǫ <

p

F (u),

∀u ∈ U.

(17)

By(14)it follows that there exists withprobability one anumber

l

0 ∈

Nsu h that there

exists apoint

x

l

₀

∈ P

l

₀

∩ U ∩ Q

. Thus, by onstru tion of

ArchiveU pdateP

Q,ǫ

, the point

x

will be dis arded from the ar hiveif it is a memberof

A

l

₀

, and neverbe added to the

ar hivefurtheron.

Nowwe onsiderthelimitbehaviorof

dist(

◦

P

P,ǫ

, A

l

)

. Itis

dist(

◦

P

Q,ǫ

, A

l

) = sup

p∈

P

Q,ǫ

◦

min

a∈A

l

kp − ak.

Let

p ∈

¯

◦

P

Q,ǫ

. For

i ∈

Nthereexistsby(14)anumber

l

i

andapoint

p

i

∈ P

l

i

∩ B

1/i

(¯

p) ∩ Q

,

where

B

δ

(p)

denotesthe openball with enter

p

and radius

δ ∈

R

+

. Sin e

lim

i→∞

p

i

= ¯

p

andsin e

p ∈

¯

◦

P

Q,ǫ

thereexistsan

i

0 ∈

Nsu hthat

p

i

∈

◦

P

Q,ǫ

forall

i ≥ i

0

. By onstru tion of

ArchiveU pdateP

Q,ǫ

, all the points

p

i

, i ≥ i

0

, will be added to the ar hive (and never

dis ardedfurtheron). Thus,wehave

dist(¯

p, A

l

) → 0

for

l → ∞

asdesired,whi h ompletes

theproof.

Remark3.11 (a) In order toobtain a ` omplete' onvergen eresult wehave postulated

some (mild) assumptions in order toguaranteethat

◦

P

Q,ǫ

= P

Q,ǫ

,whi h isin fa t an

importanttopologi al property neededfor the proof. However, if we drop the

assump-tions we an still expe tthat the interiorof

P

Q,ǫ

the `interesting' part (see Lemma

3.9)willbeapproximatedinthelimit. Tobemorepre ise,regardlessofassumptions

(A1)(A3) itholdsinthe above theoremthat

lim

l→∞

dist(

◦

P

Q,ǫ

, A

l

) = 0,

withprobability one.

(b) Note the analogy of the approa h proposed above toone approa h to approximate the

Paretofront( ompareto[13 ℄): in aseallthenondominatedsolutionswhi harefound

sofar are keptin thear hive,i.e., when

ArchiveU pdateN D(A, P ) := {x ∈ A ∪ P : y 6≺ x ∀y ∈ A ∪ P }

is used, one an show that under similar assumptions as in Theorem 3.10 an

appli ation of Algorithm1leadstoasequen eof ar hives

{A

i

}

i∈

N

,su hthat

lim

l→∞

d

H

(F (P

Q

), F (A

l

)) = 0

with probability one,

where

P

Q

denotes the Pareto set of

F

_Q

,i.e.

(15)

4 Numeri al Results

Here we demonstrate the pra ti ability of the novel ar hiver on two examples. For this,

we ompare

ArchiveU pdateP

Q,ǫ

against the ` lassi al' ar hiving strategy whi h storesall

nondominated solutions obtained during the sear h pro edure (

ArchiveU pdateN D

). To

obtain a fair omparison of the two ar hiverswe have de ided to take a random sear h

operatorforthegenerationpro ess(thesamesequen eofpointsforallsettings).

4.1 Example 1

Firstwe onsider theMOP suggestedbyTanaka([15℄):

F :

R

2 _→

R

2 _,

_{F (x}

1 , x

2 ) = (x

1 , x

2 )

(18) where

C

1 (x) = x

2

1 + x

2

2 − 1 − 0.1 cos(16 arctan(x

1 /x

2 )) ≥ 0

C

2 (x) = (x

1 − 0.5)

2 + (x

2 − 0.5)

2 ≤ 0.5

Figure4showstwo omparisonsfor

N = 10, 000

and

N = 100, 000

pointswithin

Q = [0, π]

2

asdomain

3

,indi atingthatthemethodis apableofndingallapproximatesolutions.

4.2 Example 2

Next,we onsiderthefollowingMOPproposed in[9℄:

F :

R

2 _→

R

2 F (x

1 , x

2 ) =

(x

1 − t

1 (c + 2a) + a)

2 + (x

2 − t

2 b)

2 (x

1 − t

1 (c + 2a) − a)

2 + (x

2 − t

2 b)

2 ,

(19) where

t

1 =

sgn

(x

1 ) min

|x

1 | − a − c/2

2a + c

, 1

, t

2 =

sgn

(x

2 ) min

|x

2 | − b/2

b

, 1

.

TheParetoset onsistsofnine onne ted omponentsoflength

a

withidenti alimages. We

have hosenthe values

a = 0.5

,

b = c = 5

,

ǫ = (0.1, 0.1)

, thedomain

Q = [−20, 20]

2

, and

N = 10, 000

randomly hosenpointswithin

Q

. Figures 5and 6displaytwotypi alresults

in parameterspa eand imagespa e respe tively. Seemingly, theapproximationqualityof

the Pareto set obtained by the limitset of

ArchiveU pdateP

Q,ǫ

is better than by the one

obtained by

ArchiveU pdateN D

, measured in the Hausdor sense. This example should

indi ate that it anbeadvantageousto store morethan just non-dominated pointsin the

ar hive,evenwhen`only'aimingforthee ientset.

Figure7showsaresultwith

N = 100, 000

Q

,andallother

values hosenasabove.

3

Totintoourframework,we onsiderinfa tthe( ompa t)domain

Q

′

_{:= [0, π]}

2 ∩ {x ∈

R

n

_{: C}

1 (x) ≥

(16)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0

0.2

0.4

0.6

0.8

1

1.2

1.4 x

1 = f

1 x

2 = f

2

(a)

N

= 10, 000

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0

0.2

0.4

0.6

0.8

1

1.2

1.4 x

1 = f

1 x

2 = f

2

(17)

−8

−6

−4

−2

0

2

4

6

8 −6

−4

−2

0

2

4

6 x

1 x

2

(a)

ArchiveU pdateN D

−8

−6

−4

−2

0

2

4

6

8 −6

−4

−2

0

2

4

6 x

1 x

2

(18)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8 f

₁

f

2

(19)

−8

−6

−4

−2

0

2

4

6

8 −6

−4

−2

0

2

4

6 x

1 x

2

(a)Parameterspa e

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8 f

1 f

2

(b)Imagespa e

N = 100, 000

(20)

4.3 Example 3

Finally,we onsidertheprodu tionmodelproposedin [12℄:

f

1 , f

2 :

R

n

_→

R,

f

1 (x) =

n

X

j=1

x

j

,

f

2 (x) = 1 −

n

Y

j=1

(1 − w

j

(x

j

)),

(20) where

w

j

(z) =

(

0.01 · exp(−(

20 z

)

2.5 ₎

for

j = 1, 2

0.01 · exp(−

z

15 )

for

3 ≤ j ≤ n

Thetwoobje tivefun tions havetobeinterpretedasfollows.

f

1

representsthesumof

the additional ost ne essaryfor amore reliable produ tion of

n

items. These items are

neededforthe ompositionofa ertainprodu t. Thefun tion

f

2

des ribesthetotalfailure

ratefortheprodu tionofthis omposedprodu t.

Here we have hosen

n = 5

,

Q = [0, 40]

n

, and

ǫ = (0.1, 0.001)

whi h orresponds to

10

per entof one ostunit for oneitem(

ǫ

1

),and to

0.1

per ent ofthe totalfailurerate (

ǫ

2

).

Figure8displaysoneresultfor

N = 1e6

Q

andwhen hoosing

ArchiveU pdateP

Q,ǫ

forthe ar hiver. In this ase atotal of

47, 369

elementsare stored in

thenal ar hive(i.e., almost

5

per entof all onsidered points),whi hshowstheneedfor

asuitablesele tionme hanismforthear hiverleadingto aredu ednumberof elementsin

thear hive.

5 Con lusion and Future Work

We have proposed and investigated anovel ar hiving strategy for sto hasti sear h

algo-rithms whi hallowsundermild assumptionsonthegenerationpro essto approximate

theset

P

Q,ǫ

whi h ontainsall

ǫ

-e ientsolutionswithin a ompa tdomain

Q

. We have

proventhe onvergen eofthealgorithmtowardthissetintheprobabilisti sense,andhave

giventwoexamplesindi atingtheusefulnessoftheapproa h.

Sin etheset ofapproximate solutionsformsan

n

-dimensional obje t,asuitablenite size

representation of

P

Q,ǫ

and therelated ar hiving strategy are of majorinterest for further

investigations.

A knoledgements

(21)

(a)Parameterspa e(proje tion)

0

20

40

60

80

100

120

140

160

180

200

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05 f

1 f

2

(b)Imagespa e

N = 1e6

(22)

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