Convergence of Stochastic Search Algorithms to Finite Size Pareto Set Approximations

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Size Pareto Set Approximations

Oliver Schuetze, Marco Laumanns, Carlos A. Coello Coello, Michael Dellnitz,

El-Ghazali Talbi

To cite this version:

Oliver Schuetze, Marco Laumanns, Carlos A. Coello Coello, Michael Dellnitz, El-Ghazali Talbi.

Con-vergence of Stochastic Search Algorithms to Finite Size Pareto Set Approximations. [Research Report]

RR-6063, INRIA. 2006. �inria-00119255v3�

inria-00119255, version 3 - 13 Dec 2006

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Thèmes COM et COG et SYM et NUM et BIO

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Convergence of Stochastic Search Algorithms to

Finite Size Pareto Set Approximations

Oliver Schütze

1

, Marco Laumanns

2

, Carlos A. Coello Coello

3

,

Michael Dellnitz

4

, and El-ghazali Talbi

1

1

INRIA Futurs, LIFL, CNRS Bât M3, Cité Scientifique

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

2

Institute for Operations Research, ETH Zurich, 8092 Zurich, Switzerland

e-mail: laumanns@ifor.math.ethz.ch

3

CINVESTAV-IPN, Electrical Engineering Department

e-mail: ccoello@cs.cinvestav.mx

4

University of Paderborn

e-mail: dellnitz@upb.de

N° 6063

November 2006

Unité de recherche INRIA Futurs

Parc Club Orsay Université, ZAC des Vignes,

4, rue Jacques Monod, 91893 ORSAY Cedex (France)

Téléphone : +33 1 72 92 59 00 — Télécopie : +33 1 60 19 66 08

Size Pareto Set Approximations

OliverS hütze

1

, Mar o Laumanns

2

, Carlos A. CoelloCoello

3

,

Mi hael Dellnitz

4

, and El-ghazaliTalbi

1

1

INRIAFuturs, LIFL, CNRS Bât M3,Cité S ientique

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

2

Institute for Operations Resear h, ETHZuri h,8092 Zuri h,Switzerland

e-mail: laumannsifor.math.ethz. h

3

CINVESTAV-IPN, Ele tri alEngineering Department

e-mail: oello s. investav.mx

4

University of Paderborn

e-mail: dellnitzupb.de

ThèmesCOMetCOGet SYMetNUM etBIOSystèmes ommuni antsetSystèmes

ognitifsetSystèmessymboliquesetSystèmesnumériquesetSystèmesbiologiques

ProjetsApi set Opéra

Rapportdere her he n°6063November200622pages

Abstra t: Inthis workwestudy the onvergen e of generi sto hasti sear h algorithms

towardtheParetosetof ontinuousmulti-obje tiveoptimizationproblems. Thefo usison

obtaining anite approximation that should apture the entire solutionset in asuitable

sense, whi h will be dened using the on ept of

ǫ

-dominan e. Under mild assumptions

about the pro ess to generate new andidate solutions, the limit approximation set will

be determined entirely by the ar hiving strategy. We investigate two dierent ar hiving

strategies whi h lead to a dierent limit behavior of the algorithms, yielding bounds on

the obtained approximation quality as well as on the ardinality of the resulting Pareto

set approximation. Finally, wedemonstrate thepotential forapossiblehybridizationof a

given sto hasti sear h algorithm with aparti ular lo al sear h strategy multi-obje tive

ontinuationmethodsbyshowingthatthe on eptof

ǫ

-dominan e anbeintegratedinto

thisapproa hin asuitableway.

Key-words: multi-obje tive optimization, onvergen e,

ǫ

-dominan e, sto hasti sear h

1 Introdu tion

A ommon goal in multi-obje tive optimization is to identify the set of Pareto-optimal

solutions(the e ientset) and itsimage in obje tivespa e, thePareto front(the e ient

frontier). Ex eptforspe ial ases,wheretheParetosetisniteorrepresentablebyanite

olle tionof linesegments(su hasin multi-obje tivelinearprogramming),it isingeneral

notpra ti abletodeterminetheentireParetoset. Instead,asuitableapproximation on ept

isneeded.

Various approximation on eptsbasedon

ǫ

-e ien y aregivenin [4℄. Asmostofthem

dealwithinnitesets,theyarenotpra ti alforourpurposeofprodu ingand maintaining

arepresentativesubsetofnitesize. Usingdis rete

ǫ

-approximationsoftheParetosetwas

suggestedsimultaneouslyby[1℄,[7℄, and[9℄. Thegeneralideaisthat ea hPareto-optimal

pointisapproximatelydominatedbysomepointoftheapproximationset.

Despite theexisten eofsuitable approximation on epts,investigationsonthe

onver-gen e of parti ular algorithms towards su h approximation sets, that is, their ability to

obtainasuitableParetosetapproximationinthelimit,haveremainedrare. Severalstudies,

su h as[2,8℄, onsideronlythe onvergen eto theentirePareto set,orto a ertainsubset

without onsidering theapproximationquality.

In[6℄theissueof onvergen etowardsanite-sizeParetosetapproximationwasnally

addressedforageneral lassofiterativesear halgorithms. Twoar hivingalgorithms were

proposed that provably maintain a nite-size approximation of all points ever generated

during thesear hpro ess. Thisled tothe laimthatthese ar hivingstrategieswill ensure

onvergen etoaParetosetapproximationofgivenqualityforanyiterativesear halgorithm

thatfullls ertainmildassumptionsaboutthepro esstogeneratenewsear hpoints. While

this laimholdstriviallyin the aseofdis rete(ordis retized)sear hspa es,itsextension

tothe ontinuous aseisnotstraightforward. Considerationofdis retizedmodels,however,

anleadtoproblemswhen, e.g.,usingmemeti strategies(metaheuristi sear halgorithms

mixedwithlo al sear hstrategieswhi hitselfusestepsize ontrol).

Thegoalofthispaperistoestablish onvergen eresultswithrespe ttoniteParetoset

approximationsforsto hasti multi-obje tiveoptimizationalgorithmsworkingin ontinuous

domains. Westartby onsideringtherstar hivingstrategyfrom[6℄andprove onvergen e

withprobabilityone toan

ǫ

-approximate Pareto set in the limit. Then weproposeanew

ar hiving strategy that additionallyensures that all elements of the limit set are

Pareto-optimalpointsitself. Forbothstrategieswegivebounds ontheapproximationqualityand

onthe ardinalityofthelimitsolutionset.

2 Ba kground

Inthefollowingwe onsider ontinuousun onstrainedmulti-obje tiveoptimizationproblems

min

x∈

R

n

{F (x)},

wherethefun tion

F

isdenedas theve toroftheobje tivefun tions

F :

R

n

→

R

k

_,

_{F (x) = (f}

1

(x), . . . , f

k

(x)),

andwhereea h

f

i

:

R

n

→

Ris ontinuous.

Denition2.1 (a) Let

v, w

∈

R

k

. Then theve tor

v

islessthan

w

(

v <

p

w

),if

v

i

< w

i

for all

i

∈ {1, . . . , k}

. The relation

≤

p

isdenedanalogously.

(b) A ve tor

y

∈

R

n

is dominatedby a ve tor

x

∈

R

n

(in short:

x

≺ y

) with respe t to

(MOP ) if

F (x)

≤

p

F (y)

and

F (x)

6= F (y)

(i.e. there existsa

j

∈ {1, . . . , k}

su hthat

f

j

(x) < f

j

(y)

),else

y

is allednon-dominated by

x

.

( ) Apoint

x

∈

R

n

is alledParetooptimaloraParetopointifthereisno

y

∈

R

n

whi h

dominates

x

.

(d) A point

x

∈

R

n

isweaklyParetooptimalif theredoesnotexistanother point

y

∈

R

n

su hthat

F (y) <

p

F (x)

.

In the following we will dene a weaker on ept of dominan e, so- alled (absolute)

ǫ

-dominan e,whi hwill beusedforourfurther studies.

Denition2.2 Let

ǫ = (ǫ

1

, . . . , ǫ

k

)

∈

R

k

+

and

x, y

∈

R

n

.

x

is said to

ǫ

-dominate

y

(in

short:

x

≺

ǫ

y

)with respe t to(MOP ) if

(i)

f

i

(x)

− ǫ

i

≤ f

i

(y)

∀i = 1, . . . , k

,and

(ii)

f

j

(x)

− ǫ

j

< f

j

(y)

for atleastone

j

∈ {1, . . . , k}

.

Wehavetoemphasizethatthe

ǫ

-dominan erelationunlikethe' lassi al'onedenedabove

is nottransitive,i.e., if

x

≺

ǫ

y

and

y

≺

ǫ

z

itdoesnotfollowthat

x

≺

ǫ

z

,but it follows

that

x

≺

2ǫ

z

. Thisfa t willbeusedinlater onsiderationsaswellasthefollowing: if

x

≺ y

and

y

≺

ǫ

z

itfollowsthat

x

≺

ǫ

z

.

Denition2.3 Let

ǫ

∈

R

k

+

.

(a) A set

F

ǫ

⊂

R

n

is alledan

ǫ

-approximateParetoset of(MOP ) ifevery point

x

∈

R

n

is

ǫ

-dominatedbyatleastone

f

∈ F

ǫ

,i.e.

∀x ∈

R

n

_:

∃f ∈ F

ǫ

:

f

≺

ǫ

x

(b) A set

F

∗

ǫ

⊂

R

n

is alledan

ǫ

-Pareto set if

F

∗

ǫ

isan

ǫ

-approximate Pareto set and if

every point

f

∈ F

∗

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 Further,let

B

δ

(x

0

) :=

{x ∈

R

n

_:

_{kx − x}

0

k < δ}

betheopenballwith enter

x

0

∈

R

n

andradius

δ

∈

R

+

.

Algorithm1givesaframeworkofageneri sto hasti multi-obje tiveoptimization

algo-rithm,whi hwillbe onsideredinthiswork. Theorem2.4statesa onvergen eresultwhi h

is loselyrelatedtothepresentwork,butwhi hleadsingeneraltounboundedar hivesizes.

Theorem2.4 [11 ℄LetanMOP

F :

R

n

→

R

k

begiven,where

F

is ontinuous,let

Q

⊂

R

n

be ompa t. Further,letthere beno weak Pareto pointin

Q

\P

Q

(where

P

Q

denotes the set

ofPareto pointsof

F





Q

),and

∀x ∈ Q

and

∀δ > 0 :

P (

∃l ∈

N

: P

l

∩ B

δ

(x)

∩ Q 6= ∅) = 1

(1)

Then anappli ation of Algorithm1,where allnon-dominatedpointsarekept,i.e.,

ArchiveU pdate(P, A) :=

{x ∈ P ∪ A : y 6≺ x ∀y ∈ P ∪ A},

generatesasequen eof ar hives

{A

i

}

i∈

N

,su hthat

lim

i→∞

d

(F (P

Q

), F (A

i

)) = 0

with probability one,

where

d(

·, ·)

denotes theHausdordistan e.

An ar hiving s heme to maintainan ar hiveof nite size wasre ently proposed in [3℄.

Newar hivemembersarerequiredtohaveadistan eofatleastapres ribedvalueof

ǫ

from

all urrent ar hive members, unless they dominate (and hen e repla e) a urrent ar hive

member. The subsequent proofof onvergen e (inprobability)is basedon the laim that

Paretopointsthat liewithin an

ǫ

-neighborhoodofanar hivememberiniteration

i

alsodo

sointhenextiteration

i + 1

,sin ear hivemembersareonlydeleted whensubstitutedbya

dominatingalternative. Thefollowingsimpleexampleshows,however,thatthis laimdoes

nothold forany

L

p

normasthe hosendistan emetri . Considertheproblem

max

x∈

R

subje ttothe onstraints

x

2

≤ 1 −

p1 − (x

1

− 1)

2

,

x

1

, x

2

∈ [0, 1],

sothattheobje tivefun tionsaretheproje tionstothe

i

-th oordinateandthe onstraint

denesthePareto set withextremepoints

(1, 0)

and

(0, 1)

. Let

A

i

=

{(0.5, 0), (0, 0.5)}

be

thear hiveatiteration

i

,whi his an

ǫ

-approximation

1

with

ǫ = 0.5

: Forallpointsin the

Paretoset,inparti ularforthepoint

(0.25, 0.25)

,thedistan eto eitherar hivememberis

notlargerthan

0.5

. Nowletthenewpoints

(1, 0)

and

(0, 1)

begenerated,whi hdominate,

and hen e repla e, bothar hivemembers. The new ar hive

{(1, 0), (0, 1)}

is only a

0.75

-approximation,asthedistan eofpoint

(0.25, 0.25)

toeitherar hivememberis

0.75

assuming

the maximumnorm and ertainly greaterthan

0.5

in all reasonabledistan e metri s. At

leastforobtainingan

ǫ

-Paretoset intheobje tivespa e, thisproblem anbeover omeby

using the

ǫ

-dominan e insteadof adistan e metri for dening theapproximationquality

aswellas forthear hiveupdatingstrategy,asproposedin[6℄.

Further,thenextexampleshowsthatwe anrunintotroublewhenusinganelitistar hiving

strategyasproposed in [3℄ in ase Fisnot inje tive: for agiven

ǫ > 0

let

f

ǫ

be asshown

inFigure1. Thatis,let

f

ǫ

havetwoisolatedglobalminima

m

1

and

m

2

with

m

1

< m

2

and

with

d(m

1

, m

2

) > ǫ

. Dene

F := (f

ǫ

, f

ǫ

+ C)

, where

C

∈

Ris a onstant. Ifthe domain

is e.g. hosenas

A := [m

1

− ǫ, m

2

+ ǫ]

, the Pareto set of the resultingMOP is given by

P = {m

1

, m

2

}

. However,sin ethe probability tond apoint

p

2

∈ A

whi h hasthe same

obje tivevalues

F (p

2

) = F (p

1

)

ofagivenpoint

p

1

∈ A

iszerointheunderlying setting,it

followsthattheset ofnondominatedpointsofagivenpopulation onsistswithprobability

oneofonesinglepoint. Thus,an

ǫ

-approximation aningeneralnotbeobtainedwhenonly

nondominatedpointsarestoredinthear hive.

3 The Algorithms

In the following we investigate two dierent strategies for the ar hiving of the solutions

foundbythealgorithmleadingto dierentlimitbehaviorsof thesequen eof ar hives

(un-der ertainadditional onditions).

First,weassumethattheentries of

ǫ

∈

R

k

+

are 'small',andthusthat itis su ientto

obtainan

ǫ

-approximate Paretoset. Forthis, we onsider thear hivingstrategyproposed

in [6℄, heregivenasAlgorithm2. It omputesthesubsequentar hive

A

ofagivenar hive

A

0

, a population

P

, and an

ǫ

∈

R

k

+

. Using this strategy, thesequen e of ar hives has a

limitbehaviordes ribedin Theorem3.2. Toshowthis,weneed rstthefollowingobvious

but ru ialpropertyof thear hivingstrategy.

1

SeeAppendixfor the denition. Notethat this on eptof

ǫ

-e ien yisnot  asthe

ǫ

-dominan e

−1

0

2

ε

0

f

_ε

Figure1: Exampleofafun tion

f

ǫ

withtwoisolatedglobalminima

m

1

= 0

and

m

2

= 2ǫ

.

Algorithm2

A := ArchiveU pdate1

ǫ

(P, A

0

)

1:

A := A

0

2: for all

p

∈ P

do

3: if

∃a ∈ A : a ≺

ǫ/3

p

then

4: CONTINUE

⊲

do notexe utelines

6



11

5: endif 6: forall

a

∈ A

do 7: if

p

≺ a

then 8:

A := A

\{a}

9: endif 10: endfor 11:

A := A

∪ {p}

12: endfor

Lemma3.1 Let

A

0

, P

⊂

R

n

benite sets,

ǫ

∈

R

k

+

,and

A := ArchiveUpdate1

ǫ

(P, A

0

)

. Thenthe following holds:

∀x ∈ P ∪ A

0

:

∃a ∈ A : a ≺

ǫ/3

x.

Proof: Roughlyspeaking,thestatementfollowssin epoints

a

areonlydis ardedfromthe

ar hiveifinturnanotherpoint

p

with

p

≺ a

isinserted(this'repla ement'isgiveninlines

7,8and 11in Algorithm 2). Tobemorepre ise, let

P =

{p

1

, p

2

, . . . , p

l

}, l ∈

N. Without

lossof generalitywe assume that all points

p

i

are onsidered in this ordering (i.e., in the

for-loopin line2ofAlgorithm2). There aretwo aseswehavetodistinguish.

CaseA:

x

∈ A

0

. Dene

p

′

0

:= x

and

p

′

i

:=



p

i

if

p

i

'repla es'

p

′

i−1

p

′

i−1

else

,

i = 1, . . . , l.

Itholdsthat

p

′

l

∈ A

andeither

p

′

l

= x

or

p

′

l

≺ x

(duetothetransitivityof

≺

). Inboth ases

itis

p

′

l

≺

ǫ/3

x

.

CaseB:

x

∈ P

. Let

x = p

j

, j

∈ {1, . . . , l}

. After the

j

-thiterationof theouter for-loopin

Algorithm2thereexistsanelement

a

j

∈ A

with

a

j

≺

ǫ/3

p

j

(line3resp. line11ofAlgorithm

2). Dene

p

′

j

:= a

j

and

p

′

i

, i = j + 1, . . . , l

,asabove. It followsthat

p

′

l

∈ A

and

p

′

l

≺

ǫ/3

x

as laimed.

Theorem3.2 Letan MOP

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

(2)

Then an appli ation ofAlgorithm 1,where

ArchiveU pdate1

ǫ

is usedtoupdate the ar hive,

leadstoasequen eof ar hives

A

l

, l

∈

N,wherethe following holds:

(a) Thereexistswithprobability onea

l

0

∈

Nsu hthat

A

l

isan

ǫ

-approximateParetoset

for all

l

≥ l

0

.

(b) Assumethereexistsa

l

0

∈

Nsu hthat

A

l

0

isan

ǫ

-approximate Pareto set. Then

A

l

= A

l

0

,

∀l ≥ l

0

.

Proof:

(a) Sin e

Q

is ompa t and

F

is ontinuousitfollowsthat

F





_Q

is uniformly ontinuous.

Hen e for

ǫ/3

∈

R

k

+

thereexists a

δ > 0

su hthat

x

≺

ǫ/3

y

∀x, y ∈ Q

with

kx − yk < δ.

(3) Dene

G :=

[

p∈P

Q

G

is an open over of

P

Q

. Sin e

P

Q

is ompa t it follows  due to the theorem of

Heine-Borelthat thereexistsanite sub over

S :=

s

[

i=1

B

δ

(p

i

)

⊃ P

Q

,

p

i

∈ P

Q

, i = 1, . . . , s.

By(2) it followsthat there exist with probabilityone

s

numbers

l

1

, . . . , l

s

∈

Nsu h

that ea h

B

δ

(p

i

)

∩ Q, i = 1, . . . , s

, gets 'visited'by

Generate

after

l

i

iterationsteps.

That is,

P

l

i

, i = 1, . . . , s

, ontainswith probability oneapoint

b

i

∈ B

δ

(p

i

)

∩ Q

, and

thus,

A

l

i

ontainswith probability oneave tor

d

i

with

d

i

≺

ǫ/3

b

i

. By onstru tion

of

ArchiveU pdate1

ǫ

thereexists forall

l

≥ l

i

withprobabilityonea

d

l

i

∈ A

l

su hthat

d

l

i

≺

ǫ/3

b

i

(seeLemma1). Set

l

0

:= max

{l

1

, . . . , l

s

}

.

Now let

x

∈ Q

. For

x

there exists a

p

∈ P

Q

su h that

F (p)

≤

p

F (x)

and sin e

S

is

a overof

P

Q

there existsan

i

∈ {1, . . . , s}

with

p

∈ B

δ

(p

i

)

. Let

l

0

, b

i

, and

d

l

i

be as

des ribed aboveandlet

l

≥ l

0

. Sin e

b

i

and

p

areinside

B

δ

(p

i

)

itfollowsby(3) that

b

i

≺

ǫ/3

p

i

and

p

i

≺

ǫ/3

p

. Hen e wehavewithprobabilityone:

d

l

i

≺

ǫ/3

b

i

≺

ǫ/3

p

i

≺

ǫ/3

p,

l

≥ l

i

.

Thus,wehavethat

d

l

i

≺

ǫ

x, l

≥ l

0

,withprobabilityoneasdesired.

(b) Thisfollowsimmediatelybythe onstru tionofAr hiveUpdate1

ǫ

(tobemorepre ise,

bylines35ofAlgorithm2).

Remarks3.3 (a) Assumption(2 )isthe ru ialparttoobtainthe onvergen e. For

gen-eral

ǫ

andgeneral

F

itis ertainlynotpossibletopostulateless. Given axed

ǫ

∈

R

k

+

itwouldinprin iple besu ienttorequire ondition(2 )onlyforthe

δ

whi h isgiven

in the proofabove aswellasfor nitely many ve tors

x

∈ Q

. However, thisis nearly

impossibleto he kinpra tise.

(b) Herewehave usedthe absolute

ǫ

-dominan e. If

0

6∈ f

i

(P

Q

), i = 1, . . . , k

,alternatively

the relative

ǫ

-dominan easin[6℄ anbeusedyielding similarresults.

( ) Wehaverestri tedthedomaintoa ompa tsubsetoftheR

n

. Thefollowing(a ademi )

example showsthat we anrun intotroubleif

Q

isnot ompa t: onsider the MOP

F :

R

+

→

R

2

F (x) = (

_{−x, −}

1

x

)

In this ase, the Pareto set isgiven by

P =

R

+

. Sin e

F (P )

isnot boundedbelow it

annotberepresentedby anitear hiveusing

ǫ

-dominan e. However,this hangesif

Next,weassumethattheentriesof

ǫ

arerelativelylarge. This anbethe asewhenthe

de isionmakerpreferstoobtainfew,widespreadsolutionsoftheMOP,orinordertobeable

to' apture'theentireParetosetwithalimitedar hive,inparti ularwhen onsideringmore

thantwoobje tives. Hen e, onvergen eoftheentriesofthesequen eofar hivestowardthe

Paretoset isdesired. Forthis, weproposeto usethear hiving strategywhi h isdes ribed

inAlgorithm3. Inthefollowingwewilldis ussthelimitbehaviorofthisapproa h.

Algorithm3

A := ArchiveU pdate2

ǫ

(P, A

0

)

1:

A := A

0

2: for all

p

∈ P

do 3: if

6 ∃a ∈ A : a ≺

ǫ/3

p

then 4:

A := A

∪ {p}

5: endif 6: forall

a

∈ A

do 7: if

p

≺ a

then 8:

A := A

∪ {p}\{a}

9: endif 10: endfor 11: endfor Lemma3.4 Let

A

0

, P

⊂

R

n

benite sets,

ǫ

∈

R

k

+

,and

A := ArchiveUpdate2

ǫ

(P, A

0

)

. Thenthe following holds:

∀x ∈ P ∪ A

0

:

∃a ∈ A : a ≺

ǫ/3

x.

Proof: AnaloguetotheproofofLemma 3.1.

Theorem3.5 Let(MOP)begivenand

Q

⊂

R

n

be ompa t,andlettherebenoweakPareto

pointsin

Q

\P

Q

. Further,let

F

be inje tive and

∀x ∈ Q

and

∀δ > 0 :

P (

∃l ∈

N

: P

l

∩ B

δ

(x)

∩ Q 6= ∅) = 1

(4)

Then an appli ation ofAlgorithm 1,where

ArchiveU pdate2

ǫ

is usedtoupdate the ar hive,

leadstoasequen eof ar hives

A

l

, l

∈

N,wherethe following holds:

(a) Thereexistswithprobability onea

l

0

∈

Nsu hthat

A

l

isan

ǫ

-approximateParetoset

for all

l

≥ l

0

.

(b) Thereexistswith probability one a

l

1

∈

Nsu hthat

( ) The limitar hive

A

∞

:= lim

l→∞

A

l

isan

ǫ

-Pareto setwith probability one.

Proof:

(a) AnaloguetotheproofofTheorem3.2(a).

(b) By(a) it follows that there exists withprobabilityone a

l

0

∈

N su h that

A

l

0

isan

ǫ

-approximate Pareto set. Assume that this number

l

0

is given.

|A

l

0

|

is ertainly

nite. Furtherlet

l

≥ l

0

. By onstru tionof

ArchiveU pdate2

ǫ

thear hive

A

l

isalso

an

ǫ

-approximatePareto set. That is, furtherpointsare onlyinsertedto thear hive

ifin turn at leastonedominatedsolutionis deleted (line 8of Algorithm3). Thusit

holdsthat

|A

l+1

| ≤ |A

l

| ∀l ≥ l

0

.

Sin e onthe other hand

|A

l

| ≥ 1 ∀l ∈

N,the sequen e

{|A

l

|}

l∈

N

of themagnitudes

ofthear hivesisbounded belowandmonotoni allyde reasingand onvergesthusto

an element

N

A

∈

N. Further, sin e

|A

l

| ∈

N

, l

∈

N, there exists a

l

1

∈

Nsu h that

|A

l

| = N

A

,

∀l ≥ l

1

.

( ) By(b) it follows that there exists with probability onea

l

1

∈

Nsu h that

|A

l+1

| =

|A

l

|, ∀l ≥ l

1

. Assumethat thisnumber

l

1

isgiven. Consideranelement

a

0

∈ A

l

with

l

_{≥ l}

1

. If

a

0

∈ P

Q

itfollowsthat

a

0

∈ A

l+m

,

∀m ∈

N

,

andthusalso

a

0

∈ A

∞

. Assume

that

a

0

6∈ P

Q

. Dene

M : Q

_→

R

M (x) := max

p∈P

Q

min

i=1,...,k

(f

i

(x)

− f

i

(p))

(5)

Undertheassumptionsmadeaboveitholdsthat

M (x)

_{≥ 0 ∀x ∈ Q}

and

M (x) = 0

⇔ x ∈ P

Q

.

Let

p

0

∈ P

Q

be the argument of themaximum of

M (a

0

)

. Sin e

a

0

6∈ P

Q

and

a

0

is

no weak Pareto point it follows that

M (a

0

) > 0

and

F (p

0

) <

p

F (a

0

)

. Sin e

F

is

ontinuousthereexistsaneigborhood

U

p

0

of

p

0

su h that

F (y) <

p

F (p

0

) +

M (a

0

)

2

· (1, . . . , 1) ∀y ∈ U

p

0

,

and thus, that

F (y) <

p

F (a

0

),

∀y ∈ U

p

0

. By(4) it followsthat

Generate

generates

withprobabilityoneafternitelymanystepsapoint

b

∈ U

p

0

∩ Q

. Nowtherearetwo

beenrepla edbyanelement

˜

a

∈

R

n

su h that

b

and

a

˜

aremutuallynon-dominating

(inthis aseset

a

1

:= ˜

a

). Inboth asesthereexists a

j

∈ {1, . . . , k}

su hthat

f

j

(a

1

) < f

j

(p

0

) +

M (a

0

)

2

.

Pro eeding in ananalogouswayweobtainasequen e

{a

i

}

i∈

N

of dominatingpoints.

Sin e the sequen e

{F (a

i

)

}

i∈

N

is below bounded and

F

is inje tive it follows that

a

i

→ a

∗

∈ Q

for

i

→ ∞

.

It remains to show that

a

∗

_{∈ P}

Q

. Forthis, assumethat

a

∗

_{6∈ P}

Q

. Dene

p

∗

asthe

argumentofthemaximumof

M (a

∗

₎

. Sin e

a

∗

6∈ P

Q

and

a

∗

isnoweakParetopointit

followsthat

F (p

∗

_{) <}

p

F (a

∗

)

and

M (a

∗

_{) > 0}

. Pro eeding further asaboveweobtain

apoint

a

∗∗

andanelement

j

∈ {1, . . . , k}

su hthat

f

j

(a

∗∗

) < f

j

(p

∗

) +

M (a

∗

₎

2

≤ f

j

(p

∗

_{) +}

f

j

(a

∗

)

− f

j

(p

∗

)

2

=

f

j

(p

∗

_{) + f}

j

(a

∗

)

2

< f

j

(a

∗

₎

Thisisa ontradi tiontotheassumptionofthe onvergen eofthesequen e,andthus

itmustbethat

a

∗

∈ P

Q

∩ A

∞

. Sin e

a

0

∈ A

l

, l

≥ l

1

,was hosenarbitrarilyitfollows

that

A

∞

isa

ǫ

-Paretosetandtheproofis omplete.

4 Bounds on the Ar hive Sizes

Inthefollowingwegiveboundsonthemagnitudeofthelimitar hives

A

∞

withrespe tto

ǫ

∈

R

k

+

andthe hosenar hivingstrategy.

For this, we have to introdu e some notations: denote by

m

i

and

M

i

the minimal resp.

maximalvalueofobje tive

f

i

, i = 1, . . . , k

,inside

Q

(thesevaluesexistsin e

F

is ontinuous

and

Q

is ompa t). Further,weneed

k

-dimensionalboxes,whi h anberepresentedby a

enter

c

∈

R

k

andaradius

r

∈

R

k

+

:

B = B(c, r) =

{x ∈

R

k

_:

|x

i

− c

i

| ≤ r

i

∀i = 1, . . . , k}.

Inthefollowingweassumethat

|P

0

| = 1

,andthusalso

|A

0

| = 1

. Thelowerboundof

|A

∞

|

for

bothar hivingstrategiesisobviouslygivenby1. Forthis, onsidere.g.

f

1

= f

2

= . . . = f

k

tobea onvexfun tionwhi htakesits(unique)minimuminside

Q

. Theupperboundsfor

thedierentar hivingstrategiesarederivedseparatelyin thefollowing.

Theorem4.1 Let

m

i

= min

x∈Q

f

i

(x)

and

M

i

= max

x∈Q

f

i

(x), 1

≤ i ≤ k

, and

|A

0

| = 1

.

Then,whenusing

ArchiveUpdate1

ǫ

,thear hivesizemaintainedinAlgorithm1forall

l

∈

N

isboundedas

|A

l

| ≤













1

ǫ

m

k

X

i1,...,ik−1=1

i1>...>kk−1

k−1

Y

j=1

(M

i

j

− m

i

j

)













,

(6)

where

ǫ

m

:= min

i=1,...,k

ǫ

i

3

.

Proof: Considerasequen e

p

1

, p

2

, . . .

ofpointswhi harealla eptedby

ArchiveU pdate1

ǫ

inthisorder(i.e.,startingwith

A

0

=

{p

1

}

). Considerthe

i

-thstepandlet

A

i

=

{a

1

, . . . , a

l

}

with

l

≤ i

. Dene

B

j

:= B(F (a

j

)

− ǫ/6, ǫ/6), j = 1, . . . , l

. Using indu tive arguments

wesee that (a)allelementsin

A

i

are mutuallynon-dominating, andthat (b)theinteriors

of all the boxes

B

j

, j = 1, . . . , l

, are mutually non-interse ting. Sin e the points

a

j

are

theupperright ornersoftheboxes

B

j

andsin etheinteriorsofthese boxesaremutually

non-interse ting theminimal distan ebetweentwo points

a

j

1

and

a

j

2

, j

1

6= j

2

, is givenby

ǫ

m

(see Figure 2). Thus weare ableto bound the numberof entries in the ar hivesifwe

anboundthenumberof su hboxeswhi h anbepla edin theimage spa e.

Letus rst onsider abi-obje tivemodel (i.e.,

k = 2

), sin ein this asethe proof is

geo-metri allydes riptiveandalready apturesthebasi idea. Sin eallpoints

a

j

aremutually

non-dominating, theimages ofthese points areall lo atedon a(virtual) ontinuously

dif-ferentiable urve

c : [m

1

, M

1

]

→

R

2

u

_{7→ (u, f(u))}

(7)

where

f : [m

1

, M

1

]

→ [m

2

, M

2

]

is astri tly monotoni ally de reasing(but not ne essarily

surje tive) fun tion. Thelengthofthis urve anbeboundedasfollows:

L(c) =

Z

M

1

m

1

kc

′

_(u)

kdu =

Z

M

1

m

1

p|1|

2

₊

_|f

′

_(u)

_|

2

_du

≤

Z

M

1

m

1

1du +

Z

M

1

m

1

|f

′

_(u)

|du =

Z

M

1

m

1

1du

₋

Z

M

1

m

1

f

′

_(u)du

≤ (M

1

− m

1

) + (M

2

− m

2

)

(8)

Thus,for

k = 2

weseethat

|A

i

| ≤

l

_(M

1

−m

1

)+(M

2

−m

2

)

ǫ

m

m

, i

_∈

N,as laimedabove.

Nowweturnourattentionto thegeneral ase,i.e. let

k

≥ 2

begiven. Dene

K := [m

1

, M

1

]

× . . . [m

k−1

, M

k−1

],

K

(i)

:= [m

1

, M

1

]

× . . . × [m

i−1

, M

i−1

]

× [m

i+1

, M

i+1

]

× . . . × [m

k−1

, M

k−1

],

and

u

(i)

:= (u

1

, . . . , u

i−1

, u

i+1

, . . . , u

k−1

), i = 1, . . . , k

− 1.

(9)

Inanalogytothebi-obje tive ase,theimagesoftheelementsofthear hivesarelo atedin

thegraphofamap

Φ

whi his hara terizedasfollows:

Φ : K

_→

R

k

Φ(u

i

, . . . , u

k−1

) = (u

1

, . . . , u

k−1

, f (u

1

, . . . , u

k−1

)),

(10)

where

f : K

→ [m

k

, M

k

]

isasu ientlysmoothfun tion satisfyingthemonotoni ity

on-ditions

∂f

∂u

i

u < 0,

∀u ∈ K

and

∀i = 1, . . . , k − 1

. Then,the

(k

− 1)

-dimensionalvolumeof

V ol(Φ) =

Z

K

p||∇f||

2

_{+ 1du =}

Z

K

s

 ∂f

∂u

1



2

+ . . . +



_∂f

∂u

k−1



2

+ 1du

≤

Z

K









∂f

∂u

1









du + . . . +

Z

K









∂f

∂u

k−1









du +

Z

K

1du

=

k−1

X

i=1

Z

K

(i)

Z

M

i

m

i









∂f

∂u

i









du

i

!

du

(i)

!

+

Z

K

1du

=

k−1

X

i=1

Z

K

(i)

−

Z

M

i

m

i

∂f

∂u

i

du

i

!

du

(i)

!

+

Z

K

1du

≤

k

X

i1,...,ik−1=1

i1>...>kk−1

k−1

Y

j=1

(M

i

j

− m

i

j

)

(11)

Thisbound of thevolume leadsdire tlyto thebound of the ardinalityof thear hivesas

statedabovewhi h on ludestheproof.

Figure 2: The entries

a

i

of ea h ar hive lie on a (virtual) urve

c

. Sin e the boxes

B

i

(withupperright orners

F (a

i

)

)aremutuallynon-interse ting,it followsthattheminimal

distan eoftwoentriesisgivenby

ǫ

m

.

Remarks4.2 (a) Sin e the onsidertations on the 'dominating map' (10 )hold also for

the Pareto front,the obtainedboundson the ar hivesizearetight.

(b) As des ribed above,

ǫ

m

is the minimal distan e between the images of two distin t

with

d(F (a), y)

≤ ∆

,where

d(

·, ·)

denotesthemaximumnormand

∆ := max

i

(M

i

−m

i

)

(e.g., when

ǫ

istoo large or the Pareto front is 'at'). Thus, following [10℄, the set

F (A

∞

)

an be viewed as an

ǫ

m

-uniform

d

∆

-representation

2

of the Pareto front (see

Appendixforthedenition). Thehugevalueof

∆

infa t,thelargestpossiblevalue

may beunsatisfying for ertainappli ations,andthusit ould beinterestingtosear h

for ar hivingstrategies whi h generate su h representations with pres ribed(smaller)

values of

∆

.

Theorem4.3 Let

m

i

= min

x∈Q

f

i

(x)

and

M

i

= max

x∈Q

f

i

(x), 1

≤ i ≤ k

, and

|A

0

| = 1

.

Then, whenusing

ArchiveUpdate2

ǫ

,the ar hivesizemaintainedin Algorithm1is bounded

for all

l

∈

Nas

|A

l

| ≤

k

Y

i=1



3

M

i

− m

i

ǫ

i



.

(12)

Proof: We an onsider thepro ess of in ludingsolutionsinto thear hiveovertime as

a pro ess for onstru ting a dire ted graph

G

. Starting with an empty graph, we add a

newnode for ea h newsolution

p

that is addedto the ar hive

A

in line 4 orline 8 ofthe

algorithm.If

p

isaddedinline8(meaningthe onditioninline7istrue),weaddar s

(p, a)

from

p

to ea h solution

a

that isdis ardedin line8due to

p

≺ a

. Let

V

t

:=

S

1≤j≤t

A

j

be

theunionofallar hivesuptoiteration

t

and

V

′

t

⊆ V

t

thesubsetofthosear hivemembers

that havebeen added in line 4. Thus, the node set of

G

t

after iteration

t

is

V

t

, and

G

t

itself is a forest whose roots are the urrent ar hivemembers

A

t

and whose leafsare the

elemetsof

V

′

t

. Obviously,thenumberofrootsmustbesmallerthanthenumberofleafs,so

|A

t

| ≤ |V

t

′

|

.

Tobound

|V

′

t

|

,thenumberofelementsthat everenteredthear hiveinline 4,weagain

onsider the boxes

B

v

:= B(F (v)

− ǫ/6, ǫ/6)

for all

v

∈ V

′

t

. Due to line 3, a solution

p

generated in iteration

t

′

_{≤ t}

annot bea eptedin line 4if

F (p)

liesinside the box

B

v

of

any previously a epted element of

v

∈ V

′

t

, otherwise

a

≺

ǫ/3

p

for some urrent ar hive

member

a

∈ A

t

as there exists

a

∈ A

t

with

F (a)

≤ F (v)

and

v

≺

ǫ/3

p

. If

p

wasa epted

in line 4,then

F (p)

annotlie inside the box

B

v

of any subsequentlya eptedelementof

v

_{∈ V}

′

t

neither, asthis would entail

p

≺ v

. Hen e, theinteriors of theboxes

B

v

must be

mutuallynon-interse ting. Themaximumnumberofnon-interse tingboxeswithsidelength

ǫ/3

and enters

c

with

m

i

≤ c

i

≤ M

i

is

Q

k

i=1

⌈3(M

i

− m

i

)/ǫ

i

⌉

,thusthe laimedbound on

thear hivesizefollows.

5 Outlook: Hybridizing with Multi-Obje tive

Continua-tion Methods

Inorder to in rease theoverall omputationalperforman e, it isoften desired to ombine

the (global) sto hasti sear h algorithm with a lo al sear h strategy. In this se tion, we

2

wantto showthat in the underlying ontextahybridizationwith multi-obje tive

ontinu-ationmethods (e.g., [5℄,[12℄) ouldbeadvantageous sin ethe on eptof

ǫ

-dominan e an

bedire tlyintegratedintothem.

Inthefollowingwe onstru tastepsizestrategyforourpurposeandshownumeri alresults

ontwo(easy)MOPs,indi atingthepossiblebenetofsu h ahybridization.

The basi idea of multi-obje tive ontinuationmethods is, roughly speaking, to move

along the set of (lo al) Pareto points. To bemorepre ise, in the ourseof the algorithm

oneis fa edwith the following setting: given apoint

x

0

∈ P |

Q

, an

ǫ

∈

R

k

+

, and asear h

dire tion

v

∈

R

n

with

kvk = 1

,thetaskistondastepsize

h

∈

R

+

su hthatforthenext

guess

y

0

= x

0

+ hv

itholds

kF (x

0

)

− F (y

0

)

k

∞

= Θǫ

m

,

(13)

where

Θ

∈ (0, 1)

isasafetyfa tor. In ase

F

isLips hitz ontinuousthere exists an

L

≥ 0

su h that

kF (x) − F (y)k ≤ Lkx − yk, ∀x, y ∈ Q.

(14)

TheLips hitz onstantaround

x

0

anbeestimatedby

L

x

0

:=

kDF (x

0

)

k

∞

= max

i=1,...,k

k∇f

i

(x

0

)

k

1

.

Combining(13)and(14),using

kx

0

− y

0

k = h

,andassumingthat

h

issu ientlysmall,we

obtainthefollowingestimation

h

≈

Θm

_L

ǫ

x

0

(15)

Notethatthisestimationonlyholdsforsmallvaluesof

ǫ

m

sin ein theother ase

h

willbe

toolarge,andthus

L

x

0

annotserveasasuitableLips hitzestimation.

5.1 Example 1

Inordertounderstandthepossibleimpa tofthedis ussionmadeaboveonthe ontinuation

methods,werstapplythestepsize ontrolonana ademi example(seealso[11℄):

F :

R

2

→

R

2

F (x) =

(x

1

− 1)

4

_{+ (x}

2

− 1)

4

(x

1

+ 1)

2

+ (x

2

+ 1)

2

!

(16)

TheParetosetofMOP (16)isgivenby

P =



λ



−1

−1



+ (1

_{− λ)}



1

1



: λ

_{∈ [0, 1]}



.

Figure 3 shows two dierent dis retizations of

P

. In Figure 3 (a) the Pareto set is

approximatedbypoints

x

i

, i = 1, . . . , N

,whi h arepla edequidistantin parameterspa e:

x

i

=



−1

−1



+

2i

N



1/

√

2

1/

√

2



.

Next,theParetosetwasdis retizedusingtheadaptivestepsize ontrolwhi hisproposed

above:

x

0

=



−1

−1



,

x

i+1

= x

i

+ h

i



1/

√

2

1/

√

2



,

where

h

i

is taken from (15)and

v

i

= (1/

√

2, 1/

√

2)

T

was hosen asthe sear h dire tion.

Figure3(b)showsthedis retizationpoints

x

i

for

ǫ

m

= 1

and

Θ = 0.99

yieldingasatisfying

distributionofthesolutionsontheParetofronta ordingtothevalueof

ǫ

m

.

5.2 Example 2

Nextwe onsider thefollowingMOP:

f

1

, f

2

:

R

n

→

R

f

i

(x) =

n

X

j=1

j6=i

(x

j

− a

i

j

)

2

+ (x

i

− a

i

i

)

4

,

(17) where

a

1

=

(1, 1, 1, 1, . . .)

∈

R

n

a

2

=

(

−1, −1, −1, −1, . . .)

∈

R

n

InFigures4and5somenumeri alresultsarepresented,wherewehaveusedthe

ontin-uationmethodproposedin [12℄. Tobemorepre ise, we haveapplied thestepsize ontrol

onthedistan ebetweenthe urrentsolutionandthepredi tor,sin ethispointmainly

de-terminesthedistan eoftwosolutions.

Figure4showstheresultfor

n = 3

and

ǫ

m

= 2

. Intotal,23solutionswereobtained. This

tsquitewellwiththebound inTheorem4.1,whi hisgivenby

|A

i

| ≤

 (25 − 0) + (25 − 0)

2



= 25.

Note that the points have not been stored a ording to one of the ar hiving strategies

proposedabove. Inthat ase,manysolutionsdependingontheinsertionorderingwould

havebeendis arded.

6 Con lusion and Future Work

Wehaveproposedgeneri sto hasti sear halgorithms forobtaining

ǫ

-approximatePareto

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

1

x

2

0

5

10

15

20

25

30

35

0

5

10

15

20

25

30

35

f

1

f

2

(a)xedstepsize

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

₁

x

2

0

5

10

15

20

25

30

35

0

5

10

15

20

25

30

35

f

₁

f

2

(b)adaptivestepsize

Figure 3: Dis retizations of the Pareto set of MOP (16) with (a) xed step size and (b)

adaptivestepsize ontrol.

limit. Wehavepresenteda onvergen eresultforthesealgorithms,and havegivenbounds

onthe ardinalityofthe orrespondingar hives.

Forfuturework,therearealotofinterestingtopi swhi h anbeaddressedtoadvan ethe

presentwork. One oulde.g. onsider thespeedofthe onvergen e,inparti ularwhenthe

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

x

1

x

2

x

3

0

5

10

15

20

25

0

5

10

15

20

25

f

1

f

2

Figure4: Resultofthe ontinuationmethod withstepsize ontrolonMOP(17)for

n = 3

inparameterspa e(left)and imagespa e(right).

0

10

20

30

40

50

60

70

80

90

0

10

20

30

40

50

60

70

80

90

100

f

1

f

2

5

10

15

20

25

30

10

15

20

25

30

35

40

f

1

f

2

Figure5: Resultofthe ontinuationmethodwithstepsize ontrolonMOP(17)for

n = 20

inimagespa e: allsolutions(left)andzoom(right).

apply this theoreti al framework in sear h for the development of fastand reliable

multi-obje tiveoptimization algorithms.

7 Appendix

Inthefollowingwestatesomedenitions whi h are usedin Theorem 2.4and Remark 4.2

Denition7.1 Let

u

∈

R

n

and

A, B

⊂

R

n

. The semi-distan e dist

(

·, ·)

andthe Hausdor

distan e

d(

·, ·)

are denedasfollows:

(a) dist

(u, A) := inf

v∈A

ku − vk

(b) dist

(B, A) := sup

u∈B

dist

(u, A)

( )

d(A, B) := max

{

dist

(A, B),

dist

(B, A)

}

Denition7.2 Let

ǫ > 0

andlet

D

⊂ Z

be adis reteset.

D

is alled a

d

ǫ

-representation

of

Z

iffor any

z

∈ Z

,thereexists

y

∈ D

su hthat

d(z, y)

≤ ǫ

.

Denition7.3 Let

Z

⊂

R

n

be any setand let

D

bea

d

ǫ

-representation of

Z

. Then

D

is

alleda

δ

-uniform

d

ǫ

-representation if

min

x,y∈D,x6=y

d(x, y)

≥ δ.

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[2℄ T.Hanne. Onthe onvergen e of multiobje tive evolutionaryalgorithms. EuropeanJournal

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