Anytime Pareto Local Search

27 Pages·2015·0.97 MB·English

Thisisapre-printversionofthearticle:J.Dubois-Lacoste,M.López-Ibáñez,andT.Stützle.AnytimeParetoLocalSearch.
EuropeanJournalofOperationalResearch,243(2):369–385,2015,doi:10.1016/j.ejor.2014.10.062
Anytime Pareto Local Search
JérémieDubois–Lacoste∗,ManuelLópez–Ibáñez,ThomasStützle
IRIDIA,CoDE,UniversitéLibredeBruxelles,50Av.F.Roosevelt,1050Brussels,Belgium
Abstract
ParetoLocalSearch(PLS)isasimpleandeffectivelocalsearchmethodfortacklingmulti-objective
combinatorialoptimizationproblems. Itisalsoacrucialcomponentofmanystate-of-the-artalgorithms
forsuchproblems. However,PLSmaybenotveryeffectivewhenterminatedbeforecompletion. Inother
words, PLS has poor anytime behavior. In this paper, we study the effect that various PLS algorithmic
components have on its anytime behavior. We show that the anytime behavior of PLS can be greatly
improvedbyusingalternativealgorithmiccomponents.WealsoproposeDynagrid,adynamicdiscretiza-
tionoftheobjectivespacethathelpsPLStoconvergefastertoagoodapproximationoftheParetofront
andcontinue toimprove itif moretime isavailable. We performa detailedempirical evaluationofthe
newproposalsonthebi-objectivetravelingsalesmanproblemandthebi-objectivequadraticassignment
problem. Our results demonstrate that the new PLS variants not only have significantly better anytime
behavior than the original PLS, but also may obtain better results for longer computation time or upon
completion.
Keywords: Paretolocalsearch,anytimeoptimization,multi-objectiveoptimization,travelingsalesman
problem,quadraticassignmentproblem
1. Introduction
Optimization problems are ubiquitous in our society and of crucial importance in numerous fields
of high social, environmental or economic relevance. In particular, many relevant situations involve
problems that are evaluated according to several criteria. These criteria are often conflicting and there
is no solution that is best for all of them at the same time. In this paper, we consider problems where
no information is given about the decision maker’s preferences. In this case, it is common to evaluate
solutionsbymeansofParetodominance. ThegoalistofindthesetofParetooptimalsolutions(thosenot
dominatedbyanyotherfeasiblesolution),oratleastacloseapproximationtoit. Thedecisionmakercan
thenchooseaposteriorithepreferredone.
We focus on algorithms to tackle multi-objective combinatorial optimization problems (MCOPs),
manyofwhichareproblemscomputationallyintractableor,moreformally,NP-hard[14]. ParetoLocal
Search(PLS)[41]isaheuristicalgorithmfortacklingNP-hardMCOPsintheParetosense. Itisanex-
tensionofiterativeimprovementalgorithmsforsingle-objectiveproblems[19]tothemulti-objectivecase
withthegoaltoapproximatethePareto-optimalset. PLScanproduce,uponcompletion,high-qualityap-
proximationswhenusedasastand-alonealgorithm[38,41]. PLSisalsoacrucialcomponentofhybrid
algorithms, where PLS is used to improve upon a set of good solutions that have been obtained by ap-
proaches,forexample,basedonscalarizations. Examplesofsuchhybridalgorithmscanbefoundforthe
∗Correspondingauthor.
Emailaddresses:[email protected](JérémieDubois–Lacoste),
[email protected](ManuelLópez–Ibáñez),[email protected](ThomasStützle)
Thisisapre-printversionofthearticle:J.Dubois-Lacoste,M.López-Ibáñez,andT.Stützle.AnytimeParetoLocalSearch.
EuropeanJournalofOperationalResearch,243(2):369–385,2015,doi:10.1016/j.ejor.2014.10.062
multi-objective traveling salesman problem [33, 34] with two and three objectives, various bi-objective
permutationflowshopproblems[7],andthebi-objectivemulti-dimensionalknapsackproblem[35].
AdisadvantageofPLSand, byextension, alimitationofthesehybridalgorithms, isthatitrequires
a possibly long time to reach high-quality approximations to the Pareto front (e.g., several hours for
largebiobjectivetravelingsalesmanproblem(bTSP)instancestackledintheliterature[34]). Moreover,
if stopped early, PLS may produce a poor approximation when compared to the final approximation
reached upon completion. For this reason, we say that PLS has a poor anytime behavior. By contrast,
an algorithm with good anytime behavior aims to deliver the best possible result at any time during its
executionbyquicklyreachinghigh-qualitysolutionsandbycontinuouslyimprovingthem[29,45].
OurgoalinthispaperistostudyandimprovetheanytimebehaviorofPLS.Inapreliminarywork[8],
weempiricallyexaminedtheimpactthatalgorithmiccomponentsofPLShaveonitsanytimebehavior,
andproposedalternativevariantsthatimproveitsanytimebehaviorforthebTSP.Inthispaper,weextend
thatpreliminaryworkinfourways.First,werepeattheexperimentsreportedinourpreliminarystudybut
consideringlongercomputationtimelimits,whichallowsustostudythepotentialfinalqualityreached
by standard PLS versus the anytime PLS variants. Second, our preliminary work used only the bTSP
asabenchmarkproblem. Inthispaper,wealsoconsiderthebi-objectivequadraticassignmentproblem
(bQAP).ThebTSPandthebQAParebothfundamentalproblemsthatariseinanumberofreal-worldsitu-
ations.TheyarealsoamongthemoststudiedMCOPsinmeta-heuristicsresearch[4,5,12,23,30,33,40].
Third, we propose a dynamic grid mechanism (Dynagrid) that dynamically limits the number of unex-
plored solutions accepted by PLS. Dynagrid extends a previous static epsilon-grid mechanism [1] by
discretizing the objective space using a decrease of epsilon at run-time. This allows PLS to converge
faster to a well-spread Pareto-front approximation, while avoiding premature convergence if the initial
discretizationwasexcessivelycoarse. Dynagridimprovesboththeanytimebehaviorandthefinalquality
achieved by PLS. Finally, we dedicate Section 7 to a survey of algorithmic components from the liter-
aturethatarerelatedtothoseproposedbyusinthispaper, andwediscussindetailthesimilaritiesand
differencesbetweenthevariousproposals.
OurexperimentalanalysisexaminesscenarioswhenPLSisusedasastand-alonealgorithmandwhen
PLSispartofahybridalgorithm. Therefore, wearecertainthattheproposedanytimevariantsofPLS
furtherimprovethestate-of-the-artlocalsearchalgorithmforthebTSP[34],notonlyintermsofanytime
behavior but also on final quality after completion. Moreover, our proposals can be applied to any bi-
objectivecombinatorialproblemandtheymayalsoturnouttobeusefulforotheralgorithmsthanPLS.
This paper is organized as follows. In Section 2, we first define multi-objective optimization and
related concepts, and then we present the original PLS algorithm and its decomposition into distinct
algorithmic components. In Section 3, we discuss the poor anytime behavior of the original PLS algo-
rithm. Section4recallsthealternativealgorithmiccomponentsproposedinapreliminarywork[8]and
presentsnewresultsthatfurtherconfirmtheimprovementsinanytimebehaviorreportedearlier.Section5
proposesDynagrid,adynamicdiscretizationoftheobjectivespacethatfurtherimprovestheanytimebe-
haviorofPLS.InSection6,wecomparetheoriginalPLSwiththebestPLSvariantsobtainedfromthe
alternative algorithmic components and the objective space discretization. We deliberately chose to re-
viewtheliteraturethatisrelevanttoourstudyattheendofthepaper(Section7),sincethisallowsusto
discussinmoredetailthesimilaritiesanddifferencesbetweenourproposalsandpreviousones. Finally,
weconcludeinSection8andhighlightpromisingdirectionsforfutureresearch.
2. ParetoLocalSearch
Inthissection,wefirstintroducebasicconceptsofmulti-objectiveoptimization.Next,weexplainthe
originalPLSalgorithmandhowwedecomposeitintodifferentalgorithmiccomponents.
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2.1. Multi-ObjectiveOptimization
Many real-world problems require optimizing more than one objective. Often, these objectives are
conflictingand,therefore,thereexistsnosolutionthatisoptimalforallobjectivesatthesametime. Ifno
aprioriassumptionsuponthedecisionmaker’spreferencescanbemade, thegoaltypicallybecomesto
findasetofsolutionsthatareoptimalinthesenseofParetooptimality.
More formally, candidate solutions are evaluated according to an objective function vector f(cid:126) =
(f ,..., f ) with d objectives. Pareto dominance defines a partial order on the set of feasible solutions.
1 d
Withoutlossofgenerality,weconsiderhereonlyproblemswhereallobjectivesaretobeminimized. If
(cid:126)uand(cid:126)varevectorsinRd,(cid:126)uissaidtodominate(cid:126)v((cid:126)u ≺(cid:126)v)iff(cid:126)u (cid:44)(cid:126)vandu ≤ v, i = 1,...,d. (cid:126)uand(cid:126)vare
i i
saidtobemutuallynon-dominated iff(cid:126)u ⊀(cid:126)v,(cid:126)v ⊀ (cid:126)uand(cid:126)u (cid:44)(cid:126)v. Forsimplicity, weextendthenotionof
dominance to solutions and we say that a solution s dominates another one s(cid:48) iff f(cid:126)(s) ≺ f(cid:126)(s(cid:48)). If a set
A does not contain any solution that dominates a solution s, and does not contain any solution s’ with
f(cid:126)(s(cid:48))= f(cid:126)(s),wewriteA⊀ s.
Ifno s(cid:48) existssuchthat f(cid:126)(s(cid:48)) ≺ f(cid:126)(s), siscalledParetooptimal. ThegoalwhentacklingMCOPsin
theParetosenseistofindthesetofPareto-optimalsolutions. Theirimageintheobjectivespaceiscalled
the Pareto front. Since this goal is in many cases computationally intractable, heuristic algorithms are
usedtofindanasgoodaspossibleapproximationtotheParetofront[11].
2.2. TheOriginalParetoLocalSearchAlgorithm
PLS is a direct extension of the iterative improvement algorithm from single-objective problems to
themulti-objectivecase. PLSexplorestheneighborhoodofsolutionsinitsarchive,insertsnewsolutions
in the archive that are non-dominated, and filters the archive to remove dominated solutions. PLS was
originally proposed by Paquete et al. [41]; independently, a similar algorithm was proposed by Angel
etal.[1].
Algorithm 1 illustrates the PLS framework. The input to PLS is an initial set of solutions A that
0
are mutually non-dominated. The solutions in A are initially marked as unexplored (line 2). PLS
0
updatesanarchiveofnondominatedsolutionsA,whichisinitiallyequaltoA ,byapplyingthefollowing
0
steps. First,asolution sischosenrandomlyamongallunexploredones(selectionsteponline5)andits
neighborhood,N(s),isfullyexplored(line6).Allneighborsthatarenon-dominatedw.r.t.thesolutionsin
A(line7)areconsideredforadditiontothearchiveA. TheprocedureUpdate(line9)addsacandidate
solution s(cid:48) ∈ N(s) to the archive if s(cid:48) is not dominated by any solution in the archive and removes
allsolutionsfromthearchivethatbecomedominatedby s(cid:48). Oncetheneighborhoodof shasbeenfully
explored,sismarkedasexplored(line12).WhenAcontainsonlysolutionsthathavebeenfullyexplored,
thealgorithmstopsinaParetolocaloptimum[42],thatis,asetofnondominatedsolutionsforwhichany
neighborsolutionisdominatedorequaltoasolutionintheset.
2.3. AlgorithmicComponentsofPLS
WedecomposePLSintothreemainalgorithmiccomponentsforwhichwewilllaterdescribeandtest
alternativevariants. Weexplainthesecomponentsindetailbelow:
Selectionstep. Thiscomponentdetermineshowtoselectthenextsolutionforneighborhoodexploration.
IntheoriginalPLS,thissolutionisselecteduniformlyatrandomamongtheunexploredones.
Acceptancecriterion. Thiscomponentdescribestheconditionsunderwhichanewsolutionentersthe
archive. TheoriginalPLSacceptsallsolutionsidentifiedintheneighborhoodexplorationthatare
non-dominated.
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EuropeanJournalofOperationalResearch,243(2):369–385,2015,doi:10.1016/j.ejor.2014.10.062
Algorithm1ParetoLocalSearch
1: Input: Aninitialsetofnon-dominatedsolutionsA0
2: explored(s):=false ∀s∈A0
3: A:=A0
4: repeat
5: s:=selectrandomlyasolutionfromA0 //Selectionstep
6: foreachs(cid:48) ∈N(s)do //Neighborhoodexploration
7: if A⊀ s(cid:48)then //Acceptancecriterion
8: explored(s(cid:48)):=false
9: A:=Update(A,s(cid:48))
10: endif
11: endfor
12: explored(s):=true
13: A0 :={s∈A|explored(s)=false}
14: untilA0 =∅
15: Output: A
Neighborhoodexploration. Thiscomponentperformstheneighborhoodexplorationoftheselectedso-
lution. Inparticular,itdefinesthepartoftheneighborhoodthatwillbeexploredbeforeswitching
to a different solution. The original PLS algorithm always explores the entire neighborhood of a
solution.
In the next section, we focus on the algorithmic components of PLS, test possible variants of them
andexaminetheirimpactontheanytimebehavioroftheresultingalgorithms.
3. AnytimeBehaviorofParetoLocalSearch
PLShasanaturalstoppingcriterion,thatis,itstopswhenallsolutionsinthearchivehavebeenfully
explored. Previous works have shown that PLS can reach a high-quality approximation of the Pareto
front in many combinatorial problems if it runs until completion [34, 36]. However, PLS may take a
longtimetostop. Intheworstcase,itmaytakeanexponentialtimetoreachcompletion,sinceforsome
problemstheParetofrontmaycontainanexponentialnumberofsolutionsand,intheoriginalPLS,the
archive size is unlimited. In practical problems, runtimes of hours for typical instance sizes have been
reported[34]. Therefore,itisdesirableinsomesituationstostopPLSbeforecompletion,butstillreturn
anashigh-qualityapproximationoftheParetofrontaspossible. Ananytimealgorithmaimstoobtainas
high-qualityresultsaspossibleatanytimeduringitsexecutionandcontinuouslyimprovesthequalityof
theresultascomputationtimeincreases[29,45]. Similarly,analgorithmthatproducesbetterresultsthan
anotherforanyterminationcriterionofpracticalinterestissaidtohaveabetteranytimebehavior.
The original PLS does not make any attempt to quickly obtain a high-quality approximation of the
Paretofrontand,hence,ifstoppedearly,theapproximationoftheParetofrontmaybepoor(seeFig.1).
Thus,wecansaythatPLS,asoriginallyproposed,isnotananytimealgorithminthesensethatitdoes
notshowagoodanytimebehavior.
Asinpreviouswork[6,8],inthispaperweusethehypervolumeindicator[46]tocomparetheanytime
behavior of multi-objective optimizers by graphically plotting the evolution of the hypervolume over
time. The hypervolume computes the size of the objective space dominated by a given non-dominated
set and, thus, higher hypervolume corresponds to better quality. In the case of PLS, we compute the
hypervolume of the current archive A of non-dominated solutions. Figure 2 illustrates this analysis; it
showsthedevelopmentoftheobtainedhypervolumeovertimefortwoalgorithms. Theplainlineshows
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6.0e+07 RDNynDa,-⊀H,V*
+07
5.8e
2
f +07
5.6e
+07
5.4e
5.2e+07 5.4e+07 5.6e+07 5.8e+07 6.0e+07
f
1
Figure1: Twonon-dominatedsetsobtainedbytwoPLSvariants,theoriginalPLS(denotedbyRND,⊀,∗)andananytimePLS
variant proposed in this paper (denoted by Dyna-HV). Each set corresponds to the output of one single run, stopped after 20
seconds,onabQAPinstancewithcorrelation−0.75. ThesetobtainedbytheanytimePLSvariantcompletelydominatestheset
obtainedbytheoriginalPLS.
e
m
u
ol
v
er
p
y
H
Algorithm 1
Algorithm 2
Time
Figure2:Anexampleofthehypervolumeattainedbytwoalgorithmsovertime.Algorithm1showsagoodanytimebehavior,while
Algorithm2showsworseanytimebehavior.
analgorithmwithgoodanytimebehavior; thealgorithmproducesaquickincreaseofthequalityatthe
beginning, and then continuously improves until the end of the execution. The dotted line shows an
algorithm with a poor anytime behavior; its quality improves slowly and it is worse than the quality of
theotheralgorithmatanytime.
In the following sections, we examine how the different components of PLS affect its anytime be-
havior. In addition, we propose variants of PLS that greatly improve its anytime behavior. These new
variantsallowstoppingPLSintimesshorterbyseveralordersofmagnitudethanitsnaturalcompletion
timeandobtainmuchbetterapproximationstotheParetofront(Fig.1isatypicalexample).
4. AlternativeAlgorithmicComponentsforAnytimeOptimization
We have recently shown [8] that the anytime behavior of PLS can be largely improved by using
alternative variants for each of its algorithmic components. In particular, we proposed the optimistic
hypervolume improvement (OHI) as an alternative to random selection; and alternative variants for the
acceptancecriterionandneighborhoodexploration.Inthissection,wefirstrecalltheseproposalsindetail,
since we will use these concepts throughout the paper. Next, we report new experimental results. In
particular,weconsiderlongercomputationtimelimitsthanthoseconsideredinourpreliminarystudy[8],
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whichgivesamoreaccuratenotionofthepotentialfinalqualityreachedbystandardPLSversusthenew
anytimePLSvariants. Moreover,weincludethebQAPasanadditionalbenchmarkproblem.
Alternativeforselectionstep. PLS does not make use of any information on the current state of the
archiveintheselectionstepbutusesasimplerandomchoice. Analternativeapproachistoselect
solutionswhoseexplorationmayhavethelargestpotentialtoimprovethecurrentarchive. Given
twosolutions sand s(cid:48) (inthebiobjectivecase),wedefinetheoptimistichypervolumecontribution
(ohvc) as the potential contribution to the hypervolume of the archive by the local ideal point
definedbysands(cid:48)intheobjectivespace. Inthebi-objectivespace,wecandefineitas:
ohvc(s,s(cid:48))=(f (s)− f (s(cid:48)))·(f (s(cid:48))− f (s)). (1)
1 1 2 2
The ohvc is based on the assumption that solutions that are close in the solution space are close
in the objective space. This common assumption is known to often hold in the single objective
case. Intuitively,ifthisistrueforeachobjectiveofamulti-objectiveproblem,itwillbetrueforthe
multi-objectiveproblemitself.Thisassumptioninthemulti-objectivecaseissupportedempirically
bysomestudies[6,15,21,37,44],andtosomeextentbytheresultsofthispaper.
Hence,byexploringtheneighborhoodofagivensolution,onecanexpecttofindnewnon-dominated
solutionsintheregionbetweenthecurrentsolutionanditsclosestneighborsintheobjectivespace.
The ohvc indicator can be used to select a pair of adjacent solutions. In the case of PLS, we de-
fineanadditionalindicatorfortheselectionofasinglesolution s,calledOptimisticHypervolume
Improvement(OHI),asfollows:

OHI(s)=o22h··voochh(vvscc((ss,,sussp)i,n+fs))ohvc(s,s ) ioiffth(cid:64)(cid:64)essrsinwufp,i,se, (2)
sup inf
where s and s aretheclosestneighborsof sinthebi-objectivespacefromthecurrentarchive
sup inf
Adefinedas
s =argmin{f (s)| f (s)> f (s)} and s =argmax{f (s)| f (s)< f (s)}.
sup 2 i 2 i 2 inf 2 i 2 i 2
si∈A si∈A
Either s or s may not exist if s is the best solution for f or f , respectively (often called
sup inf 1 2
extremesolution). Insuchacase, wedefinetheOHItobetwotimestheoptimistichypervolume
contributionoftheexistingsolutioninordertoavoidastrongbiasagainstextremesolutions.Fig.3
showsagraphicalrepresentationoftheOHIindicator. TheOHIvalueisaheuristictomeasurethe
“gap”aroundsolutionsintheobjectivespace,inordertoselectforexplorationasolutionthathas
the highest potential of increasing the hypervolume. This is different from measuring the actual
hypervolume contribution of the solution [3, 24]. Computing the OHI of a set of solutions in the
bi-objectivespacecanbedoneinlineartime,ifthesetissortedwithrespecttooneobjective. For
thisreason,whenaddingorremovingsolutions,wealwayskeepthesetsorted. TheOHIcouldbe
appliedtomorethantwoobjectives,butanefficientimplementationwouldrequireadatastructure
tofindefficientlythenearestsolutionstoagivenone. Weleavethisextensiontofuturework.
AsanalternativetotherandomselectionintheoriginalPLS,denotedby(cid:104)RND(cid:105),theOHIselection,
denotedby(cid:104)OHI(cid:105),selectstheunexploredsolutionwiththehighestOHIvalue.
Alternativesforacceptancecriterion. TheoriginalPLSalgorithmacceptsanynon-dominatedsolution
for inclusion in the archive. We call this component (cid:104)⊀(cid:105) for “non-dominated”. However, differ-
ent criteria have been used in the literature, particularly more restrictive ones that avoid a large
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Figure 3: Representation of the normalized objective space. The OHI
ofsolutionsisthesumofthetwohatchedareasthatliebetweensand
itstwoclosestneighborsintheobjectivespace, sinf and ssup. TheOHI
ofextremesolutions(inblack)istwicetheareabetweenthemandtheir
closestneighbor.
number of solutions in the archive [28] (a detailed discussion of how our proposals relate to the
existingliteratureisgiveninSection7). Inparticular,acceptingonlyneighborsthatdominatethe
current solution may allow a quick convergence to the Pareto front at the price of a possible loss
of quality. We call this component (cid:104)(cid:31)(cid:105) for “dominating”. We can also switch from one rule to
another: ifasolutionthatdominatesthecurrentoneisfound,onlysuchsolutionisaccepted,andif
nodominatingsolutioncanbefound,theacceptancecriterionswitchestoacceptingsolutionsthat
arenon-dominated. Wecallthiscomponent(cid:104)(cid:31)⊀(cid:105).
Alternativesforneighborhoodexploration. PLSexploresinitsoriginalversionallneighboringsolu-
tions. Thisisalsodoneiniterativeimprovementalgorithmsthatuseabest-improvement pivoting
ruleand,hence,werefertothiswayofneighborhoodexplorationinPLSalsoasbest-improvement
andrefertoitascomponent(cid:104)∗(cid:105).Insingle-objectivelocalsearch,afirst-improvementneighborhood
exploration often leads faster to local optima [19]. In the multi-objective case, first-improvement
correspondstostoppingtheneighborhoodexplorationassoonasoneneighboringsolutioncanbe
accepted. We refer to this alternative as (cid:104)1(cid:105). In PLS with first-improvement neighborhood ex-
ploration a solution is marked as explored when a neighbor is accepted into the archive even if
there could be more acceptable neighbors if the full neighborhood were explored. We also pro-
posedswitchingfromonealternativetotheotherinthefollowingway: whenallsolutionsinthe
archivehavebeenmarkedasexploredusingthefirst-improvementrule,thealgorithmwillmarkall
solutionsasunexploredandexplorethemagainusingthebest-improvementrule. Ideally,thefirst-
improvement phase will quickly converge to a good approximation of the Pareto front, while the
best-improvementphasewilladdanyremainingneighborstothearchiveand,thus,furtherimprove
itsquality. Wedenotethisalternativewith(cid:104)1∗(cid:105).
Inwhatfollows,wedenoteeachvariantoftheoriginalPLSalgorithmbyspecifyingwhichalternative
componentsitisusing. Forinstance,PLS(cid:104)OHI,(cid:31)⊀,1∗(cid:105)denotesthevariantthat
• usesOHIfortheselectionstep;
• usesarulefortheacceptancecriterionthatswitchesfromdominatingtonon-dominated;
• uses a rule for the neighborhood exploration that switches from stopping after the first accepted
neighbortostoppingonlyafterconsideringallacceptedneighbors.
Followingthisnotation,theoriginalPLSisdenotedbyPLS(cid:104)RND,⊀,∗(cid:105).
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4.1. ExperimentalSetup
4.1.1. InitialSetstoStartPLS
Inourexperimentalanalysis,wecoverpossibleusesofPLSbyusingthreedifferentinitialconditions,
thatis,setsofsolutionsthatPLSstartsfrom.
• High-qualitysets(HQS).State-of-the-artalgorithmsforseveralproblems[7,10,33,34]usePLS
inasecondphasetorefineasetofhigh-qualitysolutionsobtainedfromafirstphase,whichoften
is based on scalarizations of the problem. To cover this usage of PLS, we consider an initial set
composed of five high-quality solutions, that is, solutions very close to the Pareto-optimal front.
These five solutions are also well-spread over the objective space, since they are obtained by an
algorithm based on scalarizations that tends to distribute solutions as evenly as possible in terms
ofthehypervolumeoftheset[6]. Wechosefiveinordertohaveoneverygoodsolutionforeach
objective(ontheextremesoftheParetofront),onesolutionthatisanequaltrade-offbetweenthe
twoextremes,andtwosolutionsthatrepresentunequaltrade-offsbetweenthetwoextremes. Since
theinitialsolutionsarenearly-optimalandwell-spread,thetaskofPLSbecomestofillthegapsbe-
tweenthesesolutions.Ahighernumberofinitialsolutionswouldmakethistaskeasier,butitwould
makehardertoassesstheanytimebehaviorsincethedifferenceintermsofhypervolumebetween
theinitialsetandthefinalonewouldbesmall. Anotherdrawbackofstartingfrommoresolutions
isthattheytaketimetobegenerated(sinceasingle-objectivealgorithmislaunchedtotackleeach
scalarizationonebyone),duringwhichtheanytimebehaviorwouldberatherpoor: stoppingdur-
ingthatphasewouldbringfewsolutions. Insomepreliminaryexperiments,weobservedthatthe
anytime behavior of different PLS variants was relatively similar, even if the absolute values are
different(sincetheinitialsethasahigherhypervolume).
• Two high-quality solutions (TS). It is possible that there is no algorithm available to solve the
scalarizedproblems,butanalgorithmisavailabletosolvethetwoindividualobjectivesofthebi-
objective problem. In this case, PLS starts from one nearly-optimal solution for each objective.
ThesetwoinitialsolutionsareontheextremesoftheParetofront,and,hence,themaintaskofPLS
istoquicklyreachthecenteroftheParetofront.
• Random solution (RS). Finally, it is possible that no algorithm is available to solve any single-
objective version of the problem, i.e., one must rely only on PLS to tackle the problem. In this
case,PLSstartsfromarandominitialsolution,whichislikelytobefarawayfromtheParetofront.
Thus,themaintaskofPLSistoquicklyreachtheParetofront,whilekeepingawidespreadsetof
solutions.
4.1.2. PerformanceAssessmentofNon-DominatedSets
Similarly to solutions that can be mutually non-dominated, different approximations to the Pareto
front can be incomparable. In fact, this is often the case when comparing the output of different algo-
rithms. Itisthereforecommontouseindicatorstomeasurethequalityoftheseapproximations. Inthis
paper,weusethehypervolume(HV)indicator,awidely-usedunaryindicatorthatisconsistentwiththe
Pareto optimality principle [46]. In the bi-objective space, the hypervolume measures the area of the
objective space that is weakly dominated by (i.e., dominated by or equal to) the image of the solutions
inanon-dominatedset. Thisareaisboundedbyareferencepointthatisworseinallobjectivesthanall
pointsinallnon-dominatedsets. Thelargeristhisarea,thebetterisanon-dominatedset.
Computingthehypervolumeofnon-dominatedsetsrequirestonormalizetheobjectivevaluesofthe
solutions in them, to avoid favoring one objective over another if the objectives do not have the same
range. For thebTSP,we foundthe lowerbound fornormalizationusing theexact Concordesolver[2],
release 03.12.19, and the upper bound for normalization by taking the worst solution value of 100000
solutionsthataresampleduniformlyatrandom.ForthebQAP,wedonotusethesameproceduresinceno
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Thisisapre-printversionofthearticle:J.Dubois-Lacoste,M.López-Ibáñez,andT.Stützle.AnytimeParetoLocalSearch.
EuropeanJournalofOperationalResearch,243(2):369–385,2015,doi:10.1016/j.ejor.2014.10.062
Table1:Cut-offtimesusedforeachproblemandtypeofinstances.
Problem Instancetype Cut-offtime(s)
bTSP Size=500 10000
Size=100,Correlation=−0.75 1200
bQAP
Size=100,Correlation=−0.5 100
exactsolverisavailablethatsolvesourinstanceswithinreasonablecomputationtime. Therefore,weran
theoriginalPLSalgorithmthreetimes,usingallinitialconditions,andwerecordthebestandtheworst
value obtained over all results. We then define the lower bound as being the best value multiplied by
0.95,andtheupperboundasbeingtheworstvaluemultipliedby1.05. Wecheckduringtheexperimental
analysis that the bounds are never exceeded by any result we obtained. All objective values are then
normalizedintotherange[1,2],andweusethecoordinates(2.1,2.1)asthereferencepointforcomputing
thehypervolumeofthenormalizedsets.
Weassessgraphicallytheanytimebehaviorbyplottingtheaveragehypervolume(over25runs)ofthe
normalizedsetsovercomputationtime.Todoso,wedefineaprioriasequenceoftimesteps,atwhichwe
normalizethecurrentnon-dominatedsetobtainedsofarbythealgorithm,wecomputeitshypervolume
and we record it. The computation time required to perform these steps is not counted in the overall
computationtimethatismeasuredandreportedinourresults, sincethesestepsarerequiredtoobserve
thebehaviorofthealgorithmsbutarenotactuallypartofthealgorithms. Byusinganexponentialscale
for the sequence of time steps (and for the plot) we can observe the behavior of the algorithms both
after short periods of time and at larger scales. More precisely, we use 100 time steps, computed as
t = (cutoff_time+1)(i/100)−1, i ∈ 1,...,100,wherecutoff_timeisacut-offtime(seeTable1),
i
determined from preliminary runs of the algorithms. In the case of the bTSP, we use a cut-off time ten
times longer than the one used in our preliminary study [8]. We also show graphically the variance of
eachalgorithmacrossthemultiplerunsbyplottingingraytheconfidenceintervalscorrespondingtoeach
curve.
4.1.3. ExperimentalBenchmark,ComputationalEnvironmentandNeighborhoodOperators
For the bTSP, we generated three instances with 500 cities. The two distance matrices of each in-
stancearegeneratedindependentlyofeachotherandcorrespondtothoseofsymmetric,EuclideanTSP
instances[6,40]. ForthebQAP,weusedtheinstancegeneratorproposedin [23]. Wegeneratedthree
instances with correlations between the flow matrices in {−0.75,−0.5,0,0.5,0.75}. The lower the cor-
relation, the higher are the run times of PLS to reach completion, since the number of non-dominated
solutionsincreasesstronglywithnegativecorrelation[38]. Ontheotherhand,PLSrequiresaveryshort
timetoterminateforinstanceswithcorrelationzeroorlarger. Hence, theseinstancesarenotusefulfor
illustratingimprovementsinanytimebehavior,andweonlypresentinthispaperresultswithcorrelation
−0.75and−0.5.
ThealgorithmsareimplementedinC++, compiledwithgcc4.4.6, andtheexperimentswererunon
asinglecoreofAMDOpteron6272CPUs,runningat2.1Ghzwitha16MBcacheunderClusterRocks
Linuxversion6/CentOS6.3,64bits.
For the bTSP, we use a neighborhood operator based on 2-exchange moves, and we use delta-
evaluationstocomputetheobjectivevalueofnewcandidatesolutionsfaster[19]. ForthebQAP,weuse
aneighborhoodoperatorbasedonexchangingtwocomponents[19]. Theinitialsolutions,whenstarting
from two or more high-quality solutions are obtained solving scalarized problems (see Section 4.1.1).
For the bTSP, the single-objective algorithm used to tackle these problems is an Iterated Local Search
algorithmbasedon3-optmoves[19]. ForthebQAP,weuseaSimulatedAnnealingalgorithm[20]. For
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Thisisapre-printversionofthearticle:J.Dubois-Lacoste,M.López-Ibáñez,andT.Stützle.AnytimeParetoLocalSearch.
EuropeanJournalofOperationalResearch,243(2):369–385,2015,doi:10.1016/j.ejor.2014.10.062
1.0 1.0 12
1.
8
HV0.6 HV0. HV10
6 1.
0.
0.2 RONHDI,⊀,⊀,*,* 0.4 RONHDI,⊀,⊀,*,* 1.08 RONHDI,⊀,⊀,*,*
1 10 100 1000 1 10 100 1000 1 10 100 1000
Seconds Seconds Seconds
0
7
0. 1
7
5 0.
60 0.6
0. 69
HV0.50 HV0.55 HV670.
0.
0 RND,⊀,* RND,⊀,* RND,⊀,*
0.4 OHI,⊀,* 45 OHI,⊀,* 65 OHI,⊀,*
0.12 1.2 12 120 1200 0. 0.12 1.2 12 120 1200 0. 0.12 1.2 12 120 1200
Seconds Seconds Seconds
0.7 71
0.
5
6 6
0. 0. 9
6
HV0.5 HV55 HV0.
0. 67
0.4 0.
RND,⊀,* 5 RND,⊀,* RND,⊀,*
0.3 OHI,⊀,* 0.4 OHI,⊀,* 0.65 OHI,⊀,*
0.1 1 10 100 0.1 1 10 100 0.1 1 10 100
Seconds Seconds Seconds
Figure4:SelectionStep: PLS(cid:104)RND,⊀,∗(cid:105)vsPLS(cid:104)OHI,⊀,∗(cid:105)foronebTSPinstance(top),andtwobQAPinstanceswithcorrela-
tion−0.75(middle)and−0.5(bottom).InitialconditionsareRS(left),TS(middle)andHQS(right).Thegrayareacorrespondingto
eachcurveshowsthe95%confidenceintervalacrossdifferentruns.
bothproblems,thesingle-objectivealgorithmsweregiventwosecondsforeachscalarization.
4.2. ExperimentalEvaluationofAlternativeComponents
WenowpresenttheexperimentalevaluationoftheanytimebehaviorofthePLSvariants.Tomakethe
presentationconcise,wepresentinthispaperplotsforonlyonebTSPinstance,onebQAPinstancewith
correlation −0.5 and another bQAP instance with correlation −0.75. Other instances show remarkably
similarresults,andtheconclusionsdrawninthepaperaretrueforallinstanceswetested. Theplotsfor
alladditionalinstancesareavailableon-lineassupplementarymaterial[9].
4.2.1. SelectionStep
WepresentinFig.4theevaluationoftheselectioncomponentthatusestheoptimistichypervolume
improvement(OHI)againsttheoriginalrandomselection(RND). Theplotsoneachrowshowtheresults
obtained with different instances: one bTSP instance (top) and two bQAP instances with correlation
−0.75 (middle) and correlation −0.5 (bottom). Each column of plots corresponds to a different initial
scenario, thatis, startingwitharandomsolution(left), twohigh-qualitysolutions(middle)andasetof
high-quality solutions (right). In most plots, PLS(cid:104)OHI(cid:105) outperforms PLS(cid:104)RND(cid:105) at any moment of the
search,sometimesbyalargegap. Thisresultindicatesthattheselectionofthemostpromisingobjective
regionsforfurtherimprovementactuallyhelpsreachingbetterresultsinlesscomputationtime. Onthe
other hand, both strategies obtain similar results on a few cases, such as the top-left plot that presents
10