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

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MOP with Stochastic Search Algorithms

Oliver Schuetze, Carlos Coello Coello, El-Ghazali Talbi

To cite this version:

Oliver Schuetze, Carlos Coello Coello, El-Ghazali Talbi. Computing the Set of Approximate Solutions of an MOP with Stochastic Search Algorithms. 2007. �inria-00157741v2�

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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

9-6399ISRNINRIA/RR--????--FR+ENG

Thèmes COM et COG et SYM et NUM et BIO

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

Oliver Schütze¹, Carlos A. Coello Coello², and El-Ghazali Talbi¹

1 INRIA Futurs, LIFL, CNRS Bât M3, Cité Scientifique e-mail: {schuetze,talbi}@lifl.fr

3CINVESTAV-IPN, Electrical Engineering Department e-mail: ccoello@cs.cinvestav.mx

N° ????

June 2007

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Unité de recherche INRIA Futurs

OliverShütze

1

, CarlosA. CoelloCoello

2

, and El-GhazaliTalbi

1 1

INRIAFuturs, LIFL, CNRS Bât M3,Cité Sientique

e-mail: {shuetze,talbi}li.fr

3

CINVESTAV-IPN, EletrialEngineering Department

e-mail: oellos.investav.mx

ThèmesCOMetCOGet SYMetNUM etBIOSystèmesommuniantsetSystèmes

ognitifsetSystèmessymboliquesetSystèmesnumériquesetSystèmesbiologiques

ProjetsApiset Opéra

Rapportdereherhe n°???? June200720pages

Abstrat: Inthisworkwedevelopaframeworkfortheapproximationof theentire setof

ǫ^-eient ^solutions^of ^a multi-objetiveoptimization problem with stohasti searh algorithms. For this,weproposethesetofinterest,investigateitstopologyandstateaonver-

generesultforageneristohastisearhalgorithmtowardthissetofinterest. Finally,we

presentsomenumerialresultsindiatingthepratiabilityofthenovelapproah.

Key-words: multi-objetiveoptimization, onvergene,ǫ^-eient^solutions,approximate solutions,stohastisearhalgorithms.

∗

Partsofthismanusriptwillbepublishedintheproeedingsofthe6thMexianInternationalConferene

onArtiialIntelligene(MICAI2007).

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Résumé: Pasderésumé

Mots-lés : Pasdemotlef

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1 Introdution

Sinethenotionofǫ^-eieny^formulti-objetiveoptimizationproblems(MOPs)hasbeen introduedmorethantwodeadesago([7℄),thisonepthasbeenstudiedandusedbymany

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

thesetofoptimalsolutions([10℄),orinordertodisretizethisset([6℄,[14℄). ǫ^-eient^solu-

tionsorapproximatesolutionshavealsobeenusedtotakleavarietyofrealworldproblems

inludingportfolioseletionproblems([17℄),aloationproblem([1℄),oraminimalostow

problem([10℄). Theexpliitomputationofsuhapproximatesolutionshasbeenaddressed

in severalstudies (e.g.,[16℄, [1℄, [3℄),in allof themsalarizationtehniqueshavebeenem-

ployed.

The sopeof this paper is to develop aframework for the approximationof the set of ǫ^-

eient solutions (denote by Eǫ⁾ ^with ^stohasti ^searh ^algorithms ^suh ^as evolutionary multi-objetive(EMO)algorithms. Thisallsforthedesignofanovelarhivingstrategyto

storethe`required'solutionsfoundbyastohastisearhproess(thoughtheinvestigation

ofthe set ofinterestwill bethemajorpartin this work). Oneinterestingfat isthat the

solutionset (the Pareto set) is inludedin Eǫ ^for âll ^(small)^values ôf ǫ^, ând ^thus ^the ^re-

sultingarhivingstrategyforEMOalgorithmsanberegardedasanalternativetoexisting

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

Theremainder ofthis paperisorganizedasfollows: in Setion 2,wegivetherequired

bakgroundfortheunderstandingofthesequel. InSetion 3,weproposeaset ofinterest,

analyzeits topology, andstate aonvergeneresult. Wepresentnumerialresultson two

examplesin Setion4and onludeinSetion 5.

2 Bakground

Inthefollowingweonsider ontinuousmulti-objetiveoptimizationproblems

xmin∈R

n{F(x)}, ^(MOP)

where the funtion F îs ^dened âs ^the ^vetor ôf ^the ôbjetive ^funtions F : ^Rⁿ →

R

k, F(x) = (f1(x), . . . , fk(x)), ând ^where êah fi : ^Rⁿ → ^R îs ontinuously dieren- tiable. Laterwewill restritthesearhtoaompatset Q⊂^Rⁿ^,^the^reader^may^think ôf

ann-dimensionalbox.

Denition2.1 (a) Letv, w∈^R^k^. ^Then ^the ^vetorv ^is^less^thanw ⁽v <pw^), ^ifvi < wi

for alli∈ {1, . . . , k}^. ^The ^relation≤p ^is^denedanalogously.

(b) y∈^Rⁿîs^dominated^byâ^pointx∈^Rⁿ⁽x≺y⁾^with^respêt^to^(MOP)îfF(x)≤pF(y)

andF(x)6=F(y)^,êlsey îsâlled nondominatedby x^.

() x∈^Rⁿ îsâlledâ^Pareto^pointîf^thereîs^no y∈^Rⁿ ^whih ^dominatesx^.

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(d) x∈ ^Rⁿ îs ^weakly ^Pareto ôptimal îf ^there ^does ^not êxist ânother ^pôint y ∈ ^Rⁿ ^suh

that F(y)<pF(x)^.

Wenowdeneaweakeroneptofdominane,alledǫ^-dominane,^whih^is^the^basis^of

theapproximationoneptusedin thisstudy.

Denition2.2 Letǫ= (ǫ1, . . . , ǫk)∈^R^k+^andx, y ∈^Rⁿ^. x^is^said^toǫ^-dominatey⁽x≺ǫy⁾

withrespetto(MOP )if F(x)−ǫ≤pF(y) ^andF(x)−ǫ6=F(y)^.

Theorem2.3([11℄) Let(MOP)begivenandq:^Rⁿ→^Rⁿ^be^dened^byq(x) =Pk

i=1αˆi∇fi(x)^,

whereαˆ îsâ^solutionôf

α∈min^R^k (

k

X

i=1

αi∇fi(x)k²2;αi ≥0, i= 1, . . . , k,

k

X

i=1

αi= 1 )

.

Then either q(x) = 0 ôr −q(x) îs â ^desent ^diretion ^for âll ôbjetive ^funtions f1, . . . , fk

in x^. ^Hene, ^eah x ^with q(x) = 0 ^fullls ^the ^rst-order ^neessary ^ondition ^for ^Pareto

optimality.

Inaseq(x)6= 0îtôbviously^follows^thatq(x)îsânâsent^diretion^forâllôbjetives. ^Next,

weneedthefollowingdistanes betweendierentsets.

Denition2.4 Letu∈^Rⁿ ^andA, B⊂^Rⁿ^. ^The semi-distane dist(·,·)^and^the ^Hausdor

distanedH(·,·)^are^dened^as^follows:

(a) dist(u, A) := inf

v∈Aku−vk

(b) dist(B, A) := sup

u∈B

dist(u, A)

() dH(A, B) := max{^dist(A, B),^dist(B, A)}

DenotebyA^the^losure^of^a^setA∈^Rⁿ^,^by

◦

Aîtsînterior,ând^by∂A=A\

◦

A^the^boundary

ofA^.

Algorithm 1givesaframework of ageneri stohasti multi-objetiveoptimization al-

gorithm,whih will be onsidered in this work. Here,Q⊂^Rⁿ ^denotes^the ^domain ^of^the

MOP,Pj ^the^andidate^set^(orpopulation)ofthegenerationproessatiterationstepj^,^and Aj ^theorrespondingarhive.

Denition2.5 A set S ⊂^Rⁿ îs âlled ^notônnetedîf ^there êxistôpen ^sets O1, O2 ^suh

that S ⊂O1∪O2^,S∩O1 6=∅^, S∩O2 6=∅^,ând S∩O1∩O2 =∅^. Ôtherwise, S îsâlled

onneted.

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Algorithm1GeneriStohastiSearhAlgorithm

1: P0⊂Q^drawn^at^random

2: A0=ArchiveU pdate(P0,∅)

3: forj= 0,1,2, . . .^do

4: Pj+1=Generate(Pj)

5: Aj+1=ArchiveU pdate(Pj+1, Aj)

6: endfor

3 The Arhiving Strategy

Inthissetionwedenethesetofinterest,investigatethetopologyofthisobjet,andnally

stateaonvergeneresult.

Denition3.1 Let ǫ ∈ ^R^k+ ^and x, y ∈ ^Rⁿ^. x ^is ^said ^to −ǫ^-dominate y ⁽x≺−ǫ y⁾ ^with

respetto(MOP) ifF(x) +ǫ≤pF(y)^andF(x) +ǫ6=F(y)^.

Thisdenitionisofourseanalogoustothe`lassial'ǫ^-dominane^relation^but^with^a^value

˜

ǫ∈^R^k−^. ^However,^we^highlightît ^here^sineît ^will^beûsed ^frequently ⁱⁿ ^this^work. ^While

theǫ^-dominaneîsâ^weakerôneptôf^dominane,−ǫ^-dominaneîsâ^strongerône.

Denition3.2 Apointx∈Qîsâlled−ǫ^weak^Pareto^pointîf^thereêxists^no^pointy∈Q

suhthatF(y) +ǫ <pF(x)^.

Nowweareabletodenethesetofinterest. Ideally,wewouldliketoobtainthe`lassial'

set

P_Q,ǫ^c :={x∈Q|∃p∈PQ :x≺ǫp}¹, ⁽¹⁾

wherePQ ^denotes^the^Pareto^set^(i.e.,^the^setôf^Paretoôptimal^solutions)ôfF

Q^. ^That^is,

everypointx∈P_Q,ǫ^c îs^`lose'^toât^leastôneêient^solution,^measuredⁱⁿôbjetive^spae.

However,sinethissetisnoteasytoathnotethattheeientsolutionsareusedinthe

denition,wewillonsideranenlargedsetofinterest(seeLemma3.9):

Denition3.3 Denote by PQ,ǫ ^the ^set ^of ^points ⁱⁿ Q⊂^Rⁿ ^whih ^are ^not −ǫ^-dominated

byany other pointin Q^,^i.e.

PQ,ǫ:={x∈Q| 6 ∃y∈Q:y≺−ǫx}² ⁽²⁾

Example3.4 (a) Figure1shows twoexamples forsetsPQ,ǫ^,ône^for ^the single-objetive ase(left), andonefor k= 2 ^(right). În^the ^rstâse^we^have PQ,ǫ= [a, b]∪[c, d]^.

1P_Q,ǫ^c ^is^losely^related^to^setE¹^onsideredⁱⁿ^[16℄.

2P_Q,ǫ^is^losely^related^to^setE⁵^onsideredⁱⁿ^[16℄.

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Figure1: Twodierentexamplesforsets PQ,ǫ^. ^Left ^fork= 1 ^andⁱⁿ^parameter^spae,^and

rightanexamplefork= 2ⁱⁿ^image^spae.

(b) Consider the MOP F :^R→^R², F(x) = ((x−1)²,(x+ 1)²)^. ^F^or ǫ= (1,1) ^andQ

suiently large, say Q= [−3,3]^, ^we ^obtain PQ = [−1,1]^and PQ,ǫ = (−2,2)^. ^Note

that the boundary ofPQ,ǫ^,^i.e. ∂PQ,ǫ =PQ,ǫ\PQ,ǫ^◦ = [−2,2]\(2,2) ={−2,2},^is^given

by −ǫ ^weak ^Pareto ^points ^whih âre ^not înluded ⁱⁿ PQ,ǫ ^(see âlso ^Lemma ^1): ^for

x1 = −2 ând x2 = 2 ît îs F(x1) = (9,1) ând F(x2) = (1,9)^. ^Sine ^there êxists ^no x∈Q^withfi(x)<0, i= 1,2^,^thereîsâlso^no^pointx∈Q^whereâllôbjetivesâre^less

than atx1 ôrx2^. ^Further,^sine F(−1) = (4,0) ând F(1) = (0,4) ^there êxist ^pôints

whih −ǫ^-dominate^these ^points, ând^theyâre^thus^not înludedⁱⁿPQ,ǫ^.

FirstwestudytheonnetednessofPQ,ǫ^. ^Theonnetednessoftheset ofinterestisan importantproperty,inpartiularwhentaklingtheproblemwithloalsearhstrategies: in

thatase,theentiresetanpossiblybedetetedwhenstartingwithonesingleapproximate

solution. Example3.4(a)showsthattheonnetednessof PQ,ǫ ânnot^beêxpetedⁱⁿâse

F(Q)îs^notônvex. ^However,ⁱⁿ^theônvexâse^the^following^holds.

Lemma3.5 Letǫ∈^R^k₊^. Îf Qîsônvexând âllfi, i= 1, . . . , k âreônvex,^then PQ,ǫ ând

F(PQ,ǫ)^are^onneted.

Proof: The proof is based on the fat that in this ase PQ ^is ^onneted ^(e.g., ^[2℄).

Assume that PQ,ǫ îs ^not ônneted, ^that îs, ^there êxist ôpen ^sets O1, O2 ⊂^Rⁿ ^suh ^that PQ,ǫ⊂O1∪O2^,PQ,ǫ∩O16=∅^,PQ,ǫ∩O26=∅^,ândPQ,ǫ∩O1∩O2=∅^. ^SinePQîsônneted,

itmustbeontainedinoneofthesesets,withoutlossofgeneralityweassumethatPQ⊂O1^.

Byassumptionthereexistsapointx∈PQ,ǫ∩O2^,^and^hene,x6∈PQ^. ^Further,^there^exists

anelementp∈PQ ^suh^thatF(p)≤pF(x)^. ^SineQ^is^onvex,^the^following^path γ: [0,1]→Q

γ(λ) =λx+ (1−λ)p ⁽³⁾

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PSfragreplaements

x p

O1 O2

γ([0,1])

PQ,ǫ

Figure2: ...

iswell-dened (i.e.,γ([0,1])⊂Q^). ^SineF îsônvexând ^by^the^hoieôfpît^holds^forâll λ∈[0,1]^:

F(γ(λ)) =F(λx+ (1−λ)p)≤λF(x) + (1−λ)F(p)≤F(x), ⁽⁴⁾

andthusthatγ(λ)∈PQ,ǫ^,^whihîsâontraditiontothehoieofO1ândO2^.

The onnetedness of F(PQ,ǫ) ^follows immediately sine images of onneted sets under ontinuousmappingsareonneted,andtheproofis omplete.

ThenextlemmadesribesthetopologyofPQ,ǫⁱⁿ^the^general^ase,^whih^will^be^needed

fortheupomingonvergeneanalysis ofthestohasti searhproess.

Lemma3.6 (a) LetQ⊂^Rⁿ ^be^ompat. ^Under ^the ^following assumptions

(A1) Letthere beno weak Pareto pointin Q\PQ^,^wherePQ ^denotes ^the ^set^of ^Pareto

pointsofF|Q^.

(A2) Lettherebeno −ǫ^weak ^Pareto ^pointⁱⁿQ\PQ,ǫ^,

(A3) DeneB:={x∈Q|∃y∈PQ:F(y) +ǫ=F(x)}^. ^LetB ⊂

◦

Q ândq(x)6= 0^for âll x∈ B^,^whereq îsâs^denedⁱⁿ ^Theorem ^{2.3 ,}

itholds:

PQ,ǫ={x∈Q| 6 ∃y∈Q:F(y) +ǫ <pF(x)}

◦

PQ,ǫ={x∈Q| 6 ∃y∈Q:F(y) +ǫ≤pF(x)}

∂PQ,ǫ={x∈Q|∃y1∈PQ :F(y1) +ǫ≤pF(x)∧ 6 ∃y2∈Q:F(y2) +ǫ <pF(x)}

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(b) Letinaddition tothe assumptionsmade abovebeq(x)6= 0∀x∈∂PQ,ǫ^. ^Then

◦

PQ,ǫ=PQ,ǫ ⁽⁶⁾

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Proof: Dene W := {x ∈ Q| 6 ∃y ∈ Q : F(y) +ǫ <p F(x)}^. ^We ^show ^the êquality PQ,ǫ = W ^by ^mutual înlusion. W ⊂ PQ,ǫ ^follows ^diretly ^by âssumption ^(A2). ^To ^see

theother inlusion assumethat there existsan x∈PQ,ǫ\W^. ^Sinex6∈W ^there êxistsân y ∈Q ^suh ^that F(y) +ǫ <p F(x)^. ^Further, ^sineF îs ôntinuous ^there êxists ^further â

neighborhoodU ^of x^suh^that F(y) +ǫ <pF(u), ∀u∈U^. ^Thus, y ^is−ǫ-dominatingall

u∈U ^(i.e.,U∩PQ,ǫ=∅^),^aontraditiontotheassumptionthat x∈PQ,ǫ^. ^Thus, ^we^have

PQ,ǫ=W ^as^laimed.

Nextweshowthat theinteriorofPQ,ǫ ^is^given^by

I:={x∈Q| 6 ∃y∈Q:F(y) +ǫ≤pF(x)}, ⁽⁷⁾

whih we doagainby mutualinlusion. Toseethat

◦

PQ,ǫ ⊂I âssume^that ^there êxistsân x∈PQ,ǫ^◦ \I^. ^Sinex6∈I ^we^have

∃y1∈Q: F(y1) +ǫ≤pF(x). ⁽⁸⁾

Sine x∈

◦

PQ,ǫ ^there êxists ^noy ∈Q ^whih −ǫ^-dominatesx^, ând ^hene,êquality ^holds ⁱⁿ

equation(8). Further,byassumption(A1) itfollowsthat y1 ^must^beⁱⁿPQ^. ^Thus,^we^an

reformulate(8)by

∃y1∈PQ: F(y1) +ǫ=F(x) ⁽⁹⁾

Sine x ∈ PQ,ǫ^◦ ^there êxists â neighborhood U˜ ôf x ^suh ^that U˜ ⊂ PQ,ǫ^◦ ^. ^Fûrther, ^sine

q(x)6= 0^byâssumption^(A1),^thereêxistsâ^pointx˜∈U˜ ^suh^thatF(˜x)>pF(x)^. ^Combining

thisand(9)weobtain

F(y1) +ǫ=F(x)<pF(˜x), ⁽¹⁰⁾

andthusy1≺−ǫx˜∈U˜ ⊂

◦

PQ,ǫ^, ^whih^is ^aontradition. Itremainstoshowthat I⊂

◦

PQ,ǫ^:

assumethereexistsanx∈I\PQ,ǫ^◦ ^. ^Sinex6∈PQ,ǫ^◦ ^there^exists^a^sequenexi∈Q\PQ,ǫ, i∈^N,

suh that limi→∞xi =x^. ^That îs, ^there êxists â^sequene yi ∈ Q^suh^that yi ≺−ǫ xi ^for

all i ∈ ^N. ^Sine âll ^the yi âre înside Q^, ^whih îs â ^bounded ^set, ^there êxists â ^subse-

quene yij, j ∈ ^N, ^and ^an y ∈ Q ^suh ^that limj→∞yij = y (Bolzano-Weierstrass). Sine

F(yij) +ǫ≤p F(xij), ∀j∈^N, ^it^follows^for ^the^limit^points ^that^also F(y) +ǫ≤p F(x)^,

whih isaontraditiontox∈I^. ^Thus, ^we^have

◦

PQ,ǫ=Iasdesired.

Fortheboundaryweobtain

∂PQ,ǫ=PQ,ǫ\

◦

PQ,ǫ

={x∈Q|∃y1∈Q:F(y1) +ǫ≤pF(x)^and 6 ∃y2∈Q:F(y2) +ǫ <pF(x)}

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Sineby(A1)thepointy1 ⁱⁿ⁽¹¹⁾^must^beⁱⁿPQ^,^we^obtain

∂PQ,ǫ={x∈Q|∃y1∈PQ:F(y1) +ǫ≤pF(x)^and 6 ∃y2∈Q:F(y2) +ǫ <pF(x)} ⁽¹²⁾