ContentslistsavailableatScienceDirect
Applied
Soft
Computing
jo u r n al hom e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / a s o c
A
learning-guided
multi-objective
evolutionary
algorithm
for
constrained
portfolio
optimization
Khin
Lwin
∗,
Rong
Qu,
Graham
Kendall
ASAPGroup,SchoolofComputerScience,UniversityofNottingham,UK
a
r
t
i
c
l
e
i
n
f
o
Articlehistory:Received2August2013
Receivedinrevisedform30March2014
Accepted11August2014
Availableonline27August2014
a
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s
t
r
a
c
t
Portfoliooptimizationinvolvestheoptimalassignmentoflimitedcapitaltodifferentavailablefinancial assetstoachieveareasonabletrade-offbetweenprofitandriskobjectives.Inthispaper,westudiedthe extendedMarkowitz’smean-varianceportfoliooptimizationmodel.Weconsideredthecardinality, quan-tity,pre-assignmentandroundlotconstraintsintheextendedmodel.Thesefourreal-worldconstraints limitthenumberofassetsinaportfolio,restricttheminimumandmaximumproportionsofassetsheldin theportfolio,requiresomespecificassetstobeincludedintheportfolioandrequiretoinvesttheassetsin unitsofacertainsizerespectively.Anefficientlearning-guidedhybridmulti-objectiveevolutionary algo-rithmisproposedtosolvetheconstrainedportfoliooptimizationproblemintheextendedmean-variance framework.Alearning-guidedsolutiongenerationstrategyisincorporatedintothemulti-objective opti-mizationprocesstopromotetheefficientconvergencebyguidingtheevolutionarysearchtowardsthe promisingregionsofthesearchspace.Theproposedalgorithmiscomparedagainstfourexisting state-of-the-artmulti-objectiveevolutionaryalgorithms,namelyNon-dominatedSortingGeneticAlgorithm (NSGA-II),StrengthParetoEvolutionaryAlgorithm(SPEA-2),ParetoEnvelope-basedSelectionAlgorithm (PESA-II)andParetoArchivedEvolutionStrategy(PAES).Computationalresultsarereportedforpublicly availableOR-librarydatasetsfromsevenmarketindicesinvolvingupto1318assets.Experimentalresults ontheconstrainedportfoliooptimizationproblemdemonstratethattheproposedalgorithmsignificantly outperformsthefourwell-knownmulti-objectiveevolutionaryalgorithmswithrespecttothequalityof obtainedefficientfrontierintheconductedexperiments.
©2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/3.0/).
1. Introduction
Portfolioselection problemisa well-studiedtopicinfinance anditisconcernedwiththeoptimalallocationofalimited capi-talamongafinitenumberofavailableriskyassets,suchasstocks, bonds,andderivativesinordertogainthepossiblehighestfuture wealth.Markowitz’smean-variancemodel [40,41]isconsidered toplayanimportantroleinthedevelopmentofModern Portfo-lioTheory.Themean-variance(MV)modelassumesthatthefuture marketoftheassetscanbecorrectlyreflectedbythehistorical mar-ketoftheassets.Itconsidersthetrade-offbetweenriskandreward inselectingefficientportfolios.Aportfolioisconsideredtobe effi-cientifitprovidesthehighestpossiblerewardforagivenriskor alternatively,ifitpresentstheleastpossibleriskforagivenlevel ofprofit.Thereward(profit)oftheportfolioismeasuredbythe
∗ Correspondingauthorat:C87,SchoolofComputerScience,Universityof
Not-tingham,NottinghamNG81BB,UnitedKingdom.Tel.:+4401158466525.
E-mailaddress:psxktl@nottingham.ac.uk(K.Lwin).
averageexpectedreturnofthoseindividualassetsintheportfolio whereastheriskismeasuredbyitscombinedtotalvariance.
WhileinvestingthecapitalwithintheMVframework,investors have two objectives: maximizing the total profit and minimiz-ingthetotal risk oftheirportfolios. Withthesetwo conflicting objectivestobeoptimizedsimultaneously,theportfolioselection problemcanbeclassifiedasamulti-objectiveoptimization prob-lem.Asinglesolutionthatoptimizesalltheconflictingobjectives simultaneouslyhardlyexistsinpractice.Instead,thereexistsaset ofacceptable‘compromise’solutionswhichareoptimalinsucha waythatnoothersolutionsaresuperiortothemwhenall objec-tivesareconsideredsimultaneously.Such solutionsarereferred toasefficientsolutions,non-dominatedsolutionsorPareto-optimal solutions.
Thecollectionofsuchefficientportfoliosconveyingthe com-promise between risk and return is called theefficient frontier or Pareto-optimalfront. The efficientfrontier helps investors to visualizetheriskandreturntrade-offcurveinatwo-dimensional graphwithriskonthehorizontalaxisandexpectedreturnonthe verticalaxis(seeFig.13).
http://dx.doi.org/10.1016/j.asoc.2014.08.026
SincetheMarkowitz’spioneeringwork,manyresearchershave pursuedstudiesforefficientalgorithms[27,29,43,52]tocompute theefficientfrontieroftheMVmodel.However,theclassicMV modelassumesaperfectmarketwhereshortsalesaredisallowed, securitiescanbetradedinany(non-negative)fractions,no limi-tationonthenumberofassetsintheportfolio,investorshaveno preferencesoverassetsandtheydonotcareaboutdifferentassets typesintheirportfolios.Inpractice,theseassumptionsare unre-alistic.Asaresult,severalextensionsandmodificationshavebeen proposedtoaddressthereal-worldconstraints.Inthispaper,we extendedthebasicMVmodeltoincludefourpracticalconstraints asfollows:
Cardinalityconstraint
Cardinalityconstraintlimitsthenumberofassets(K)that com-posetheportfolio.Veryofteninpractice,investorsprefertohave alimited numberof assetsincludedin theirportfoliosincethe managementofmanyassetsintheportfolioistediousandhard tomonitor.Theyalsointendtoreducetransactioncostsand/orto assureacertaindegreeofdiversificationbylimitingthemaximum numberofassetsintheirportfolios.
Floorandceilingconstraints
Thefloorandceilingconstraintsspecifytheminimumand max-imumlimitsontheproportionofeachassetthatcanbeheldina portfolio.Inpractice,investorsprefertoavoidexcessive admin-istrativecostsfor very smallholdings ofassets in theportfolio and/orsomeinstitutionalpoliciesrequiretomodeltheirpolicies onthelowerandupperboundsofeachassetintheportfolio.The floorandceilingconstraintisalsoknownasboundingorquantity constraints.
Pre-assignmentconstraint
Thepre-assignment constraint is usually used tomodel the investor’ssubjectivepreferences.Aninvestormayintuitivelywish somespecificassetstobeincludedintheportfolio,withits propor-tionfixedortobedetermined.
Roundlotconstraint
Roundlotconstraintrequiresthenumberofanyassetinthe portfolioto bein exact multipleof thenormal tradinglots. In practice,severalmarketsecuritiesaretradedasmultiplesof mini-mumlots.
Thesefourconstraintsstatedabovearehardinthesensethat theyhavetobesatisfiedatanytime.Inpractice,portfoliosare com-posedofmarketswithhundredstothousandsofavailableassets, andthecalculationofriskmeasuresgrowsquicklyinrelationtothe numberofassets.Byintroducingthecardinalityconstraintalone alreadytransformstheclassicquadraticoptimizationmodelinto amixed-integerquadraticprogrammingproblemwhichisan NP-hardproblem[6,47].Thereareseveralexactapproachesproposed intheliteratureforcardinalityconstrainedportfoliooptimization problem[5,6,35,47].However,alltheseworksrelaxedthe cardi-nalityconstraintasaninequalityconstraintallowingthenumber ofassetsintheportfoliotovarywithmaximumbound(K)andthe resultsshowedthattheyareabletohandlethetestproblemswith limitedsize(upto500assets).Ontheotherhand,Gulpinaretal.
[26]consideredthestrictcardinalityconstraintandcomputational resultsareperformedonasmalltestprobleminvolving98assets. WhenadditionalconstraintsareaddedtothebasicMVmodel, the problemthus becomes more complex and the exact opti-mization approaches run into difficulties to deliver solutions withinreasonable time forlargeproblemsize. Asa result, this
motivates theinvestigation of approximate algorithms suchas meta-heuristics[33]andhybridmeta-heuristics[56,45].Ingeneral, meta-heuristicscannotguarantee theoptimalityofthesolution, buttheyareefficientinfindingtheoptimalornearoptimal solu-tionsinareasonableamountoftime.
Thereexistmanystudieswhichappliedmeta-heuristicsorother techniquestosolveportfoliooptimizationproblem[21,39].The recentresearchinportfoliooptimizationproblemiswidely car-ried out by incorporation of constraints in theproblem model and/orhandlingtheproblemasamulti-objectiveone.Althoughthe portfoliooptimizationprobleminvolvestwoconflictingobjectives, manystudiesintheliterature[11,17,20,37]havebeenperformed assingleobjectivemeta-heuristicsapproacheswithaggregating functionthatcombinestwoobjectivesintoa singlescale objec-tive,andin whichtheweights arevariedtogeneratethesetof efficientsolutionsforportfolioselectionproblemswith cardinal-ityandquantityconstraints.MansiniandSperanza[38]showed thattheportfolioselectionproblemwithroundlotconstraintis anNP-completeproblemandproposedthreemixedintegerlinear programmingheuristicalgorithmstosolvetheproblem.Linand Liu[36]proposedageneticalgorithmwiththreedifferentmodels forportfolioselectionproblemswithroundlots.Changetal.[11]
andGasperoetal.[25]discussedthepre-assignmentbrieflybut hadnotaddressedtheconstraintintheirexperiments.
In recent years, many publications had discussed the port-folio optimization problems with multi-objective evolutionary algorithmsbyconsideringasubsetofthereal-worldconstraints. Diosan [22] and Mishra et al. [42] appliedseveral well-known multi-objectiveevolutionaryalgorithmstosolvetheunconstrained portfolio optimization problem. Recently, Krink et al. [34] also proposedanalgorithmcalledDEMPOinspiredbytheNSGA-II algo-rithm[19].ThedifferencebetweenNSGA-II and DEMPOis that DifferentialEvolution(DE)is usedinsteadofGeneticAlgorithm (GA)togenerate newcandidatesolutions duringtheevolution. DEMPOisappliedtosolvethebasicportfoliooptimization prob-lembasedonValue-at-Riskriskmeasureandexperimentalresults showthatDEMPOoutperformsNSGA-II.ArmananzasandLozano
[3]studiedgreedysearch,simulatedannealing(SA)andantcolony optimization(ACO)algorithmsinamulti-objectiveframeworkto solvetheportfolioselectionproblemwithcardinalityconstraints. AnagnostopoulosandMamanis[2]consideredtheextendedMV modelwithcardinality andquantityconstraintsand testedfive advancedMOEAstoinvestigatetheperformance.Thecardinality constraintconsideredin theirworkis relaxedand asa resulta portfoliocanbecomposedofanynumberofassetswithmaximum bound (K). The experimental results confirmed that all multi-objectivealgorithmsconsideredoutperformedthesingleobjective evolutionaryalgorithm.TheresultsalsoconcludedthatSPEA-II[60]
performedthebestamongthosealgorithmstested.Brankeetal.[7]
alsopresentedanenvelope-basedMOEAintegratingtheNSGA-II
[18]andthecriticallinealgorithm.Chaimetal.[12]proposedan order-basedsolutionrepresentationandconsideredthe cardinal-ityconstraintasasoftconstraintandquantityconstraintasahard constraint.Intheirwork,thecardinalityconstraintwasrelaxedand henceitwasallowedtovarythenumberofassetsintheportfolio fromtheminimumlimittothemaximumlimit.
Streichert et al. [55,54]applied a multi-objective evolution-ary algorithm(MOEA)tosolvetheportfolio selectionproblems with cardinality, floor and round lot constraints. These works studiedvariouscrossoveroperatorsadoptinghybridchromosome representationwithbinaryandrealvalues.Thishybridencoding enhancestheperformanceofthealgorithmsignificantly regard-lessofthechoiceofcrossoveroperators.Skolpadungketetal.[50]
alsostudiedtheportfolioselectionproblemswithcardinality,floor androundlotconstraintsandtestedthemwithvariousMOEAs. TheyadoptedthesamehybridencodingasStreichertetal.[55,54].
Experiments are performedonthesmall datasetcontaining 31 assetsandtheperformancemetricsshowedthatSPEA-II[60]is thebestalgorithmamongthosetested.Intheirwork,the cardi-nalityconstraintwasrelaxedandonlythemaximumcardinality constraintwasconsidered.
Fieldsendet al. [23] and Anagnostopoulos and Mamanis [1]
consideredthecardinalityconstraintasanadditionalobjectiveto beminimized.Brito and Vicente [8]reformulated the cardinal-ityconstrainedMVmodelasabi-objectiveproblem,allowingthe investorstoanalyzetheefficienttrade-offbetweenmean-variance andcardinality.Thedetailedreviewsofthemulti-objective evo-lutionary algorithms in portfolio optimization can be found in
[10,13,44,49].
Inthiswork,weproposea newlearning-guidedhybrid evo-lutionaryalgorithmforthemean-varianceportfoliooptimization problemwithinthecontextofthemulti-objectiveoptimization.We extendedtheMVmodeltoconsiderthestrictcardinality,quantity, pre-assignmentandroundlotconstraints.
Weforthefirsttimeinvestigatetheperformanceofthe learning-guided multi-objective evolutionary algorithm with external
archive (MODEwAwL) on the extended MV model with four
constraints considered. Randomly generating a new candidate solutionisveryunlikelytoachieveagood-qualitypractical solu-tionfortheconstrained portfoliooptimizationproblem.Instead, alearning-guidedsolutiongenerationschemeincorporating addi-tional problem-specific heuristics is proposed to generate a good-quality solution. The proposed algorithm contributes to enhanceanefficientconvergenceofthesearchalgorithmby con-centratingonthepromisingareasofthesearchspace.
In this study, we consider four existing well-known multi-objectiveevolutionary algorithms(MOEAs), theNon-dominated SortingGeneticAlgorithm(NSGA-II)[19],theStrengthPareto Evo-lutionaryAlgorithm(SPEA2)[60],ParetoEnvelope-basedSelection Algorithm(PESA-II)[16]and ParetoArchivedEvolutionStrategy (PAES)[30].Alargesetofsimulationexperimentshavebeen con-ductedoveranumberofinstances.Resultsdemonstratethatthe proposedalgorithmishighlyefficientintermsofbothfinding solu-tionsclosetothetruePareto-frontandgooddistributionalongthe Pareto-front.
Therestofthepaperisorganizedasfollows.Section2describes thegenericmulti-objectiveportfoliooptimization,followedbythe extended MV model considering the cardinality, quantity, pre-assignment and round lot constraints. Section3 introduces the proposedalgorithmoutliningthemaindifferencesfromthe exist-ingapproaches. Section4providesthedetailed structure ofthe proposedalgorithm.Section5discussestheanalysisofthe simula-tionresults.InSection6,conclusionandfutureworkarepresented.
2. Multi-objectiveportfoliooptimization
Multi-objective optimizationgenerallyinvolves balancingall conflicting objectives and searches for a set of compromise solutions between the objectives while satisfying the various constraints.Insuchcontext,thissetofsolutionsisknownas Pareto-optimalsolutions[18].
Inmulti-criteriavariantofportfoliooptimizationproblem,the MVmodelcanbeformalizedasabi-objectiveoptimization prob-lem.Theobjectiveistofindasetofefficientportfoliosthatmaximize returnandminimizerisksimultaneously.Inthiswork,four real-worldconstraints,cardinality,quantity,pre-assignmentandround lot,are considered(seeSection1).Mathematically,theproblem withconsideredconstraintscanbeformulatedasfollows: min f1= N
i=1 N j=1 wiwjij (1) max f2= N i=1 wii (2) subject to N i=1 wi=1 (3) N i=1 si=K, (4) wi=yi.i, i=1,...,N, yi∈Z+ (5) isi≤wi≤ıisi, i=1,...,N, (6) si≥zi, i=1,...,N (7) si,zi∈{0,1}, i=1,...,N (8)whereNisthenumberofavailableassets,iistheexpectedreturn
ofasseti(i=1,...,N),ijisthecovariancebetweenassetsiandj
(i=1,...,N;j=1,...,N),andwi(0≤wi≤1)isthedecisionvariable whichrepresentstheproportionheldofasseti.Eq.(3)definesthe budgetconstraint(allthemoneyavailableshouldbeinvested)fora feasibleportfolio.
Eq.(4)definesthecardinalityconstraintwhereKisthenumber ofinvestedassetsintheportfolioandsidenoteswhetherassetiis
investedornot.Ifsiequalstoone,assetiischosentobeinvested
andtheproportionofcapitalwiliesin[
i,ıi],where0≤i≤ıi≤1.Otherwise,assetiisnotinvestedandwiequalstozero.
In this study, we adopted the strict cardinality constraint
[3,11,37,39,55,54]and thus requiretoselectfixedKnumber of assets. Experimentalresultsfrom theliterature[11,55] showed thatwhenthecardinalityconstraintwithhighKvalueisimposed, theapproximationof theconstrained efficientfrontiertendsto approachtowardstheunconstrainedefficientfrontier(UCEF).The cardinality constrainthasbeenrelaxedinseveralrelatedworks
[2,7,50],wheretheequalityconstraintisreplacedby inequality constraint (i.e.uptoKassets canbeincludedin theportfolio). Insomeworks[12,25],thecardinalityconstraintisalternatively relaxed by specifying themaximum and minimum number of assetsthataportfoliocanhold.
Eq. (7) defines the pre-assignment constraint to fulfil the investors’ subjective requirements where the binary vector zi
denotesifassetiisinthepre-assignedsetthathastobeincludedin theportfolioornot.Eq.(5)definestheroundlotconstraintwhere yiisapositiveintegervariableand
iistheminimumlotthatcanbepurchasedforeachasset.Theinclusionofroundlotconstraint maymakeitimpossibletoexactlysatisfythebudgetconstraint(see Eq.(3))asthetotalcapitalmightnotbetheexactmultiplesofthe requiredtradinglotforvariousassets.
Theabovestatedmodelcouldbesolvedbyobtainingasetof effi-cientportfolios.Theseobtainedsolutionsareoptimalinthesense thattherearenoothersolutionsinthesolutiondomainorsearch spacethataresuperiortothemwhenallobjectivesareconsidered simultaneously[18].Thecomplete setof theseefficient portfo-liosformstheefficientfrontierthatrepresentsthebesttrade-offs betweenthemeanreturnandthevariance(risk).Inpractice,when morereal-worldconstraintsareconsidered,theefficientfrontier reducestoasmallercurve.
Inatwo-dimensionalspaceofriskandreturn,asolutionais saidtobeefficient(i.e.,Pareto-optimal)iftheredoesnotexistany
solutionbsuchthatbdominatesa[24].Solutionaisconsideredto dominatesolutionbifandonlyif:
f1(a)≤f1(b) AND f2(a)>f2(b) OR
f2(a)≥f2(b) AND f1(a)<f1(b)
Theultimategoalsinamulti-objectiveportfoliooptimization problemaretofindasetofsolutionsascloseaspossibletothe Pareto-optimalfrontandtofindagooddistributionofsolutions alongtheParetofront.Oncetheefficientfrontierisobtained,the decisionmakerdeterminestheportfoliobasedontheinvestor’srisk preference.Hence,thediversityofthesolutionsalongtheefficient frontieris importantforthedecisionmakernottomisscertain trade-offportfolioswhichhe/shemightbeinterested.
3. Learning-guidedmulti-objectiveevolutionaryalgorithm
(MODEwAwL)
Themulti-objective portfoliooptimization problembecomes toocomplextosolvebynumericalmethodswhenthose practi-calconstraintsreflectinginvestors’preferencesand/orinstitutional tradingrules areconsidered. Overthe last twodecades, multi-objective evolutionary algorithms (MOEAs) have received a significantamountofattentionanddemonstratedtheir effective-nessandefficiencyinapproximatingthePareto-optimalfront[13]. DEMO[46]isoneoftherecentalgorithmswhichcombinesthe advantagesofDE[53]withthemechanismsofPareto-basedsorting andcrowdingdistancesorting[19].Ithadbeensuccessfullytested onthecarefullydesignedtestfunctions(ZDT)introducedin[59]. TheprocedureoftheDEMOisdescribedinFig.1.DEMOmaintainsa populationofindividuals,whereeachrepresentsapotential solu-tiontotheoptimizationproblem.Duringtheevolution,itallows itspopulationcapacityexpandinordertoaddnewlyfound non-dominatedsolutions(seeFig.1,lines3–9).Hence,itenablesthe newlyfoundnon-dominatedsolutionstoimmediatelytakepartin thegenerationofthesubsequentcandidatesolutions.Thisfeature ofDEMOpromotesfastconvergencetowardsthetrueParetofront. Ineachgeneration,ifthepopulationexceeds thesizelimit,itis sortedbasedonthenon-dominationandcrowdingdistancemetrics
[19]inordertoidentifythoseindividualstobetruncated.Itthus aimstomaintainagooddistributionofnon-dominatedportfolios. Inthiswork,weproposealearning-guidedmulti-objective evo-lutionary algorithm (MODEwAwL) for the constrained portfolio optimization.Theproposedalgorithmadoptsanewapproachto extendgenericDEMOschemetosolvetheconstrainedportfolio optimizationproblem.Themaindifferencesofourapproachwith respecttotheDEMOschemeintheliteraturecanbeoutlinedas follows:
Fig.1.TheprocedureofDEMO[46].
Fig.2.TheprocedureoftheproposedMODEwAwL.
1.Asecondarypopulation(i.e.anexternalarchive)isintroducedto storethewellspreadnon-dominatedsolutionsfound through-outtheevolution(seeSection4.9).
2.Alearningmechanismisproposedtoextracttheimportant fea-turesfromtheefficientsolutionsfoundthroughouttheevolution (seeSection4.4).
3.Anefficientsolutiongenerationschemeutilizingthelearning mechanism,problemspecificheuristicsandeffective direction-basedsearchmethodsisproposedtoguidethesearchtowards thepromisingsearchspace(seeSection4.5).
TheproposedMODEwAwLusethearchivetoextractthe impor-tantfeaturesofnon-dominatedsolutions.Incorporatinglearning mechanismandpriorproblem-specificknowledgeexploitationin theevolutionprocessallowstheproposedMODEwAwLtogenerate promisingoffspringsolutions.TheproposedMODEwAwLthusaims topromoteconvergencebyconcentratingonthepromisingregions ofthesearchspace.Thepseudocodeoftheproposedalgorithmis describedinFig.2.
4. TheproposedMODEwAwL
4.1. Notation Let
A thearchivemaintainingthesetofnon-dominated port-folio(s)
CR thecrossoverprobabilityfordifferentialevolution F thescalingfactorfordifferentialevolution
K thenumberofassetsinaportfolio,i.e.thecardinality L thenumberofassetsinthepre-assignmentset
M themaximumsizeofthearchive
N thenumberofavailableassets
NP thenumberofindividualsinthepopulation P listofportfoliosinthepopulation
ci theconcentrationofithassetinthearchive pi theithportfoliointhepopulation
wi theproportionofcapitalinvestedintheithasset
i theminimumtradinglotoftheithasset i thelowerboundontheproportionoftheithassetıi theupperboundontheproportionoftheithasset r[x1,x2] randomrealvaluebetweenx1andx2,bothinclusive R[x1,x2] randomintegervaluebetweenx1andx2,bothinclusive
si=
1 if the ith (i=1,...,N) asset is chosen 0 otherwise
zi=
1 if ith asset is in pre−assigned set 0 otherwise
4.2. Solutionrepresentationandencoding
Inoursolutionrepresentation,twovectorsofsizeNareusedto defineaportfoliop:abinaryvectorsi,i=1,...,Ndenotingwhether
assetiisincludedintheportfolio,andareal-valuevectorwi,i=1, ...,Nrepresentingtheproportionsofthecapitalinvestedinthe assets.Someexistingresearches[2,50,55,54]adoptsimilar encod-ingtodefineaportfolio.Whenthecardinalityandpre-assignment constraintsareconsidered,theintroductionofbinaryvariablessi
inthemulti-objectiveportfoliomodelenhancestheevaluationof thealgorithm.
4.3. Initialpopulationgeneration
Togenerateaninitialpopulation,Kdifferentindexes(including allassetsinthepre-assignmentsubset)arerandomlyselectedand proportionsareassignedtothoseselectedassetsrandomly.Ifthe generatedportfolioviolatesthebudgetand/orquantityconstraints, suchsolutioniscorrectedbytheconstrainthandlingtechniques providedinSection4.6.Hence,allgeneratedsolutionsinthetrial populationarefeasible.
4.4. Learningmechanism
At each generation, the distribution of assets from non-dominatedsolutionsintheexternalarchiveisobservedtoidentify thepromisingassets.Theconcentrationscoreofeach assetciis
calculatedbycountingitsoccurrencesinthearchivedividedby archivesize.
ci=
|A| j=1si,j|A| .
Thenewsolutionstobegeneratedareencouragedtocompose withthoseassetsbyexploitingtheknowledgeobtainedthroughout theevolutiontodirectthesearchtowardsthepromisingsearch space.Theproposedlearningmechanismiscomputationallycheap asitonlyusesasingleupdateateachgeneration.Asimilarform ofscoringfunctionhasbeenusedasoneofthecomponentsinthe trade-offstudiesbySmithetal.[51].
4.5. Candidategeneration
Oneofthefactorstoconsiderindesigningtheportfoliomodel intheproposedMODEwAwListofindaneffectivewayto gener-ateoffspring.Weaimtofindeffectiveandefficientschemewitha goodbalancebetweentheexploitationandexploration.Thenew solutionisgeneratedbytwophases:theselectionofassetsfroma universeofNavailableassetsandtheallocationofcapitaltothose selectedassets.TheideapresentedhereistouseDEfor explor-ingtherealdecisionvariablesandexploitlearningmechanismand
problemspecificheuristicsdescribedbelowtoselectthepromising assetsinthenewsolution.
Theinformationabouttheconcentrationof theassetsin the non-dominatedportfoliosinthearchiveisexploitedinselecting thepromisingassetsforthenewcandidateportfolio.Hencethe assetsarerankedaccordingtotheirconcentrationsinthearchive non-dominatedsolutions.Theassetswhichscoregreaterthanzero areconsideredtobepromisingones.Thehigherthescoreofthe asset,thehigheritschancestobeincludedinthenewcandidate portfolio(seeSection4.4).
In finance literature, it is considered to be a fundamental premisetoutilizeassetsthathavelowcorrelationwitheachother. Hencetheassetswhicharelesscorrelatedtoeachotherare prefer-abletotheheavilycorrelatedassets.Itisalsocommonlybelieved thatitisbeneficialtoreducetheportfolio’sstandarddeviationof return.Intuitively,investorspreferhigherreturnassetswithless risk[28].
Inordertogenerateanewcandidatesolution,theLassetsare firstlyselectedifthepre-assignmentconstraintisconsidered.By takingintoaccountoftheabovestatedintuitivelearning,inthis work,theproposedMODEwAwLthenalternativelyusesthe fol-lowingselectionschemestofilltheremainingassets:
S1 The(K−L)assetsareselectedbyroulettewheelselection basedontheconcentrationscoreci.
S2 The(K−L)assetswhichhavethehighestconcentration scoreciareselected.
S3 The(K−L)assetswhichhavethehighestexpectedreturn valuesareselected.
S4 Therandomnassets(wheren=R[0,K−L])whichhavethe highestconcentrationscoreciareselected.Theremaining
(K−n)assetsarefilledbyselectingoneofthefollowing methods.
•Selectthoseassetswhichhavethelowestriskvalues.
•Selectthoseassetswhichhavethehighestreturnvalues(i.e.S3). •Selectthoseassetswhichhavetheleastcorrelationfromthosen
assetsalreadychosen.
By adopting the above stated selection scheme, the new candidate solution satisfies the pre-assignment and cardinality constraints.Theproportionsofthoseselectedassetsforthenew candidate solutionare assigned by usinga direction-based off-spring generation scheme where p1, p2 and p3 are randomly selectedportfoliosfromthecurrentpopulationPasfollows: W1 wi:=w3i+r[0,1]×(w1i−w2i)
W2 wi:=w3i+F×(w1i−w2i)
W3 rankp1,p2andp3bydominanceandcrowdingdistance measure(i.e.p1isthebestportfolioandp3istheworst portfolioamong three portfolios)and generate weight allocationsofcandidateportfoliobydirectingawayfrom p3andtowardsthemiddlebetweenp1andp2asfollows:
wi:= w1i+w2i 2
Thedetailedprocedureofthecandidategenerationisprovided inFig.3.Theproposedcandidategenerationmechanismintends toguidethesearchtowardpromisingdirectionbylearningfrom thereferenceassets fromthearchiveandreferenceproportions fromthecurrentpopulation.Inthisway,theproposedalgorithm convergesefficiently. Thenewcandidateportfoliois repaired if thequantityandroundlotconstraintsareviolated(seetherepair mechanisminSection4.6).
Fig.3.Theprocedureofgeneratingacandidatesolution.
4.6. Constrainthandling
Whenusing an evolutionary algorithm to solveconstrained optimizationproblems,therearevariousmethodsproposedinthe literature[15]forhandlingconstraintsinevolutionary optimiza-tion,suchaspenaltyfunctionmethod,specialrepresentationsand operators,repairmethodsand multi-objective methods.Among thosemethods,repairmethodisoneoftheeffectiveapproaches tolocatefeasiblesolutions.
Duringthepopulationsampling,each constructedindividual portfolioisrepairedifitdoesnotsatisfyallconsideredconstraints. Asdescribed in Section4.5, the newsolutiongenerated byour proposedMODEwAwLalready satisfiesthecardinality and pre-assignmentconstraints.
Hence, the following repair mechanism stated in [50,54] is applied:
1.All weights of selected asset in the candidate solution are adjustedbysettingwi=
i+(wi−i)/(wi−
i).2.Theweights arethenadjusted tothenearestroundlot level bysettingwi=wi−(wi mod
i).Theremainingamountof capitalisredistributedinsuchawaythatthelargestamountof (wi mod i)isaddedinlotofiuntilallthecapitalisspent. 4.7. SelectionschemeTheproposedMODEwAwLappliestheelitistselectionscheme basedonParetooptimality(seeFig.2).Duringtheevolution,the populationisextendedbyaddingthenewlyfoundnon-dominated solutions.Hence,ateachgeneration,thenumberofportfoliosin thecurrentpopulationwillbebetweenNPand2NP.
4.8. Truncatepopulation
Ineachgeneration,ifthenumberofportfoliosinthecurrent populationexceedsitslimitNP,itneedstoidentifythosewhich needtoberemoved.Theindividualsinthepopulationaresorted basedonthenon-dominanceandcrowdingdistancemetrics.Then thecurrentpopulationistruncatedbykeepingthebestNP individ-ualsforthenextgeneration.
4.9. Maintainingtheexternalarchive
ThemainobjectiveoftheexternalarchiveAistokeepallthe non-dominatedsolutionsencountered alongthesearchprocess. Thisapproachisadoptedinordertosaveandupdateallwellspread non-dominatedsolutionsgeneratedbythealgorithmduringthe search.
In each generation, the archive Ais updated withthe non-dominatedsolutionsfromthetrialpopulation.Thecomputational timeof maintainingthearchive increaseswiththearchivesize
[14,32,60]. The size of the archive is therefore restricted to a pre-specified value.When theexternal archive hasreachedits maximumcapacityM,thecrowdingdistancesofthesolutionsare calculatedtodeterminethemostcrowdedarchivememberswhich needtobediscarded.
5. Performanceevaluation
Inthissection,wefirstintroduce thetestproblemsand per-formancemetricsusedforevaluatingtheproposedMODEwAwL. Wethenstudytheeffectivenessofthetwocomponentsextended forMODEwAwL,i.e.theexternalarchiveandthelearning-guided solutiongenerationscheme,respectively.Finally,wecomparethe proposedMODEwAwLwithfourstate-of-the-artmulti-objective evolutionaryalgorithmsintermsoftheperformancemetrics. 5.1. Dataset
Seventestproblemsbasedonwell-knownmajormarketindices fortheportfoliooptimizationproblems fromthepublicly avail-able OR-library[4] is usedto evaluate theperformance of the algorithms.Table1showsthedetailsofthesebenchmarkindices andtheirsizes.Thefirstfivedatasets(D1–D5)builtfromweekly pricedatafromMarch1992toSeptember1997areavailableat:
http://people.brunel.ac.uk/∼mastjjb/jeb/orlib/portinfo.html. They werefirstintroducedbyChangetal.[11].
The remaining two datasets were built based on the index tracking problem and they were first introduced by Canakgoz andBeasley[9].Thesetwodatasets(D6andD7)areavailableat:
http://people.brunel.ac.uk/∼mastjjb/jeb/orlib/indtrackinfo.html. These probleminstances have been used for portfolio opti-mizationwithcardinality,quantity,pre-assignmentandroundlot constraintsinordertostudytheperformances.Allalgorithmshave beenimplementedinC#andrunonapersonalcomputerIntel(R) Core(TM)2 Duo CPU E8400 3.16GHz. The experimental results obtainedforeachalgorithmaretheaverageof20runs.
5.2. Qualityindicators
Toevaluatetheperformanceofthemulti-objective evolution-aryalgorithmsfromvariousaspects,severalperformancemetrics havebeenproposedintheliteraturewhichmainlyconsider prox-imity,diversityanddistribution.Inthisstudyweusefourwidely adoptedperformanceevaluationmetricsnamelygenerational dis-tance,invertedgenerationaldistance,diversityandhypervolume. 5.2.1. Invertedgenerationaldistance(IGD)
Theinverted generationaldistance[48] usesthetruePareto frontasareferenceandmeasuresthedistanceofeachofits ele-ments from the true Pareto front to the non-dominated front obtainedbyanalgorithm.Itismathematicallydefinedas:
IGD=
Q i=1d 2 i Q Table1TheBenchmarkinstancesfromOR-library.
Instance Origin Name Numberofassets
D1 HongKong HangSeng 31
D2 Germany DAX100 85 D3 UK FTSE100 89 D4 US S&P100 98 D5 Japan Nikkei 225 D6 US S&P500 457 D7 US Russell2000 1318
Fig.4. Effectivenessofthelearning-guidedsolutiongenerationschemeandarchive.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferred
tothewebversionofthearticle.)
whereQisthenumberofsolutionsinthetrueParetofrontand
diistheEuclideandistancebetweeneachofthesolutionandthe
nearestmemberfromthesetofnon-dominatedsolutionsfoundby
thealgorithm.Thismetricmeasuresboththediversityandthe
con-vergenceofanobtainednon-dominatedsolutionset.Thesmaller
thevalueofthismetric,theclosertheobtainedfrontistothetrue
Paretofront.
The true Paretofront for highly constrained multi-objective
portfolio optimization problem considered in this work is
unknown. We use the best known unconstrained efficient
Fig.6. PerformancecomparisonsoffivealgorithmsintermofGD,IGDandmetricsforDAX100dataset.
frontier(UCEF)providedbytheOR-library[4]asthetruePareto
frontreferenceset.Thishasbeenwidelyadoptedintheliterature. 5.2.2. Generationaldistance(GD)
Thegenerationaldistance[57]isavariantofIGD.Itmeasures howfarthesolutionsofthecomputedParetofrontobtainedbyan algorithmarefromthoseatthetrueParetofront.Thesmallervalue indicatesthatallthegenerated solutionsareonthetruePareto front.
5.2.3. Diversitymetric()
Thediversitymetric()[19]measurestheperformanceindices ofdistributionandspreadsimultaneouslyfortwo-objective opti-mizationproblems.Thediversitymetric()isdefinedasfollows: = df+dl+
|Q|−1 i=1 |di−d| df+dl+(|Q|−1)dwherediistheEuclideandistanceintheobjectivespacebetween
consecutivesolutionsintheobtainednon-dominatedfrontQ,and distheaverageofthesedistances.Theparametersdfanddlarethe
Euclideandistancebetweentheextremesolutionsandthe bound-arysolutionsoftheobtainednon-dominatedfrontQ.Thelower valueofthespread()indicatesabetterdiversity.
5.2.4. Hypervolume(HV)
Hypervolumemetric[61],alsoknownasS-metricorLebesgue measure,iswidelyrecognizedasanunaryvaluewhichisableto measurebothconvergenceanddiversity.Thismetriccalculatesthe normalizedvolumeoftheobjectivespacecoveredbytheobtained ParetosetQboundedbyareferencepointr.Therefore,highervalues arepreferable.Foreachsolutioni∈Q,ahypercubecifromsolution iandthereferencepointrismeasured.ThehypervolumeHVis calculatedas:
HV=volume(∪|Q| i=1ci)
AnaccuratecalculationofHVrequiresanormalizedobjective spaceandweusedthelinearnormalizationtechniqueproposedby Knowlesetal.[31]asfollows:
fi= fi−fimin fmax i −f min i wherefmin
i andfimaxaretheminimumandmaximumvalueofthe ithobjective.Thevalueoffmin
i andf
max
i aresetastheminimum and maximumvalueobtainedfromrunningallalgorithms.The referencepointwaschosenasr={1,0}.
5.3. Effectivenessofthelearning-guidedsolutiongenerationand archive
In this section, ourexperiments focus onthe impactof the learning-guidedsolutiongenerationmechanism.Inorderto eval-uatetheperformanceoftheMODEwAwL,wecompareitwithtwo variantsofthealgorithm:themulti-objectivedifferentialevolution (MODE)andthemulti-objectivedifferentialevolutionwitharchive (MODEwA).Fig.4showsthecomparisonsofthethreealgorithms in termsof IGD, GDand.The experimentalresultsdistinctly showthattheproposedalgorithmwiththelearning-guided solu-tiongenerationmechanismoutperformsMODEandMODEwAin mostinstances.
5.4. Theoverallperformanceevaluation
Inordertoevaluatetheoverallperformanceoftheproposed MODEwAwL, we compare it with four state-of-the-art multi-objectiveevolutionaryalgorithmsintheliterature.
•NSGA-II:theNon-dominatedSortingGeneticAlgorithmIIwas proposedby Debet al. [19]. Thealgorithm usesbinary tour-namentselectionbasedonthecrowdingdistance.It performs
Fig.7.PerformancecomparisonsoffivealgorithmsintermofGD,IGDandmetricsforFTSE100dataset.
crossoverandmutationbysimulatedbinarycrossoverand poly-nomialmutationoperators.
•SPEA2: the Strength Pareto Evolutionary Algorithm was pro-posedbyZitzleretal.[60].Thealgorithmemploysfine-grained
fitnessassignment,density estimationtechniquesandarchive truncation methods. Like NSGA-II, it uses binary tournament selection,simulatedbinarycrossoverandpolynomialmutation evolutionaryoperators.
Fig.9. PerformancecomparisonsoffivealgorithmsintermofGD,IGDandmetricsforNikkeidataset.
•PESA2:theParetoEnvelope-basedEvolutionaryAlgorithmwas proposedbyCorneetal.[16].Thealgorithmuseshyper-boxesto assignfitnessandemploysthesimulatedbinarycrossoverand polynomialmutationoperations.
•PAES:theParetoArchivedEvolutionStrategywasproposedby Knowles and Corne [30]. The algorithm uses a simple (1+1) local search evolution strategy. It maintains an archive of
non-dominatedsolutionsanditexploitsthoseParetosolutions toestimatethequalityofnewsolutions.
Inordertoensureafaircomparison,wehaveusedthesame populationsizeandarchivesize(ifapplicable)forallthealgorithms testedinthiswork.Wehavechosentorunallthealgorithmsrunfor thesamestoppingcriteria(i.e.thesamenumberofevaluations)to
Table2
Parametersettingoffivealgorithms.
Parameters MODEwAwL NSGA-II SPEA2 PESA-II PAES
Numberofpopulation(NP) 100 100 100 100 100
Numberofgeneration 1000N 1000N 1000N 1000N 1000N
Scalingfactor(F) 0.3 – – – –
Crossoverprobability(CR) 0.9 0.9 0.9 0.9 –
Crossoverdistributionindex – 20 20 20 –
Mutationprobability – 1/N 1/N 1/N 1/N
Mutationdistributionindex – 20 20 20 20
Tournamentround – – 1 – –
Numberofbisection – – – 5 5
Archivesize(M) 100 – 100 100 100
generatetheParetofront.Eachalgorithmalsousesthesame
encod-ings(seeSection4.2)andrepairmechanism(seeSection4.6)when
anewlyconstructedportfolioviolatestheconsideredconstraints. Beforetheexperimentswereperformed,parametersaretunedfor allalgorithmsusingthesmallestprobleminstance,i.e.HangSeng.
Table2showsthebestparametervaluesofthealgorithms.
5.5. Comparisonsofthealgorithms
Inthissection,wehaveperformedanumberofexperiments. TheresultsofGD,IGDandandrunningtimeofthefive algo-rithmsperformedonsevendatasetsfromOR-libraryareshownin
Figs.5–11.ForexampleinFig.11,topleftboxplotrepresentsthe performanceofeachalgorithmconsideredintermsofGDmetric, toprightboxplotrepresentstheperformanceofeach algorithm consideredintermsofIGDmetric,bottomleftshowsthe perfor-manceofeachalgorithmintermsofDiversitymetricandbottom rightboxplotdisplaysthecomputationaltimeforeachalgorithm considered.Theseresultsareobtainedfortheconstrainedportfolio optimizationproblemwithcardinalityK=10,floor
i=0.01,ceilingıi=1.0,pre-assignmentz30=1androundlot
i=0.008.Theresultsshowthatformostoftheprobleminstances,the MODEwAwLobtainsthesmallestmeanvaluesforGD,IGDand, comparedwiththeotherfouralgorithms,demonstratingthebest
performanceamongthefivealgorithms.NSGA-IIcomesatthe sec-ondandSPEA2comesatthethirdplaces.NSGA-IIandSPEA2seem tohave almostcomparable resultsfor mostprobleminstances. However,SPEA2isthemostcomputationallyexpensivealgorithm interms ofCPU time.TheresultsalsoconfirmthatPAESis the worst algorithm for theportfolio optimizationwith considered constraints.However,PAESisthesecondfastestalgorithm after MODEwAwL.Formostoftheprobleminstances,theproposed algo-rithmMODEwAwLisalsocomputationallyefficientcomparedto theothers.Fig.12showsthehypervolume(HV)calculation per-formedonsevendatasetsandforeachprobleminstance,theresults reconfirmthesuperiorityofMODEwAwLsinceitoutperformsinsix outofsevendatasets.
For illustrative purpose, the obtained efficient frontiers of the algorithms for seveninstances along with thetrue uncon-strainedefficient frontier(UCEF)are providedin Fig.13.When theproblemsizesaresmall,theParetosetobtainedbythe con-sidered algorithms is very competitivetoeach other suchthat it would behard todifferentiatevisually. Astheproblemsizes increase,theproposedalgorithmobtainedsignificantlybetter effi-cient frontier than those obtainedby other MOEAs considered in this work.Based ontheanalysis, weconclude thatthe pro-posedMODEwAwLisabletosolvelarge-scalereal-worldportfolio optimizationefficiently.TheresultsalsodemonstratethatNSGAII
Fig.12.PerformancecomparisonsoffivealgorithmsintermofHVmetric.
andSPEA2loosetheireffectivenesswhentheproblemdimension increases.
Togainanintuitiveviewofthefivealgorithmsovergenerations, weplottheGD,IGDandmetricsovergenerationsonfiveselected instancesinFig.14wheretheresultsareaveragedover20runs.The resultsconfirmthatallalgorithmsconsideredareabletoconverge andMODEwAwLisabletoconvergethefastestinmostproblem instances.
Experimentsarealsoperformedfordifferentcardinalityvalues withK=15andK=5.Theresultsaremadepubliclyaccessibleat:
http://cs.nott.ac.uk/∼ktl/results/MODEwAwL-results.pdf. On the majorityofdatasets,MODEwAwLissignificantlybetterthanthe other compared MOEAs. The experimentalresults have further demonstratedthattheproposedalgorithmisefficientforvarious searchspaceswithdifferentvaluesofK.TheproposedMODEwAwL isthusmorerobustthanthecomparedMOEAs.
Fig.13.Comparisonofefficientfrontiersforsevendatasets.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionof
Fig.14. Comparisonsofconvergenceoffivealgorithms.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthe article.)
Table3
Student’st-testresultsofdifferentalgorithmsonsevenprobleminstanceswithK=10,i=0.01,ıi=1.0,z30=1andi=0.008.
Algorithm1↔Algorithm2 HangSeng DAX100 FTSE100 S&P100 Nikkei S&P500 Russell2000
MODEwAwL↔NSGA-II ∼ + + + + + + MODEwAwL↔SPEA2 − + + + + + + MODEwAwL↔PESA-II ∼ + + + + + + MODEwAwL↔PAES + + + + + + + NSGA-II↔SPEA2 − + ∼ ∼ + + ∼ NSGA-II↔PESA-II + + + ∼ + + ∼ NSGA-II↔PAES + + + + ∼ + − SPEA2↔PESA-II + ∼ ∼ ∼ + ∼ ∼ SPEA2↔PAES + + + + − ∼ − PESA-II↔PAES + + + + − ∼ − Table4
Student’st-testresultsofdifferentalgorithmson5probleminstanceswithK=15,i=0.01,ıi=1.0,z30=1andi=0.008.
Algorithm1↔Algorithm2 HangSeng DAX100 FTSE100 S&P100 Nikkei S&P500 Russell2000
MODEwAwL↔NSGA-II ∼ + + + + + + MODEwAwL↔SPEA2 ∼ + + + + + + MODEwAwL↔PESA-II + + + + + + + MODEwAwL↔PAES + + + + + + + NSGA-II↔SPEA2 + ∼ + + + ∼ ∼ NSGA-II↔PESA-II + + + + + ∼ ∼ NSGA-II↔PAES + + + + + + ∼ SPEA2↔PESA-II + ∼ ∼ ∼ + ∼ ∼ SPEA2↔PAES + + + + − ∼ ∼ PESA-II↔PAES + + + + − ∼ ∼ Table5
Student’st-testresultsofdifferentalgorithmsonfiveprobleminstanceswithK=5,i=0.01,ıi=1.0,z30=1andi=0.008.
Algorithm1↔Algorithm2 HangSeng DAX100 FTSE100 S&P100 Nikkei S&P500 Russell2000
MODEwAwL↔NSGA-II + + + + + + + MODEwAwL↔SPEA2 + + + + + + + MODEwAwL↔PESA-II + + + + + + + MODEwAwL↔PAES + + + + + + + NSGA-II↔SPEA2 − + ∼ + + ∼ ∼ NSGA-II↔PESA-II + + ∼ + + ∼ ∼ NSGA-II↔PAES + + + + − − − SPEA2↔PESA-II + ∼ ∼ ∼ + ∼ ∼ SPEA2↔PAES + + ∼ ∼ − − − PESA-II↔PAES ∼ + ∼ + − − −
AsstatedinSection5.2.1,IGDcanprovidetheoverall
perfor-manceofanalgorithm,measuringitsconvergenceanddiversity simultaneously.WecomparetheIGDvaluesofthefivealgorithms byusingStudent’st-test[58].Thestatisticalresultsobtainedby a two-tailed t-test with 38 degrees of freedom at a 0.05 level ofsignificancearegiveninTables 3–5.Theresultof Algorithm-1↔Algorithm-2isshownas“+”,“−”,or“∼”whenAlgorithm-1 is significantlybetterthan, significantly worsethan, or statisti-callyequivalenttoAlgorithm-2, respectively.Results showthat MODEwAwLoutperformsotheralgorithmsinmostoftheproblem instancesexceptHangSengdataset.ForHangSengtestproblem, theperformanceofSPEA2outperformsMODEwAwLwhenK=10. WethereforecanconcludethattheproposedMODEwAwLhasthe bestoptimizationperformancefortheportfoliooptimization prob-lemwithconsideredconstraints.
6. Conclusionandfuturework
Inthiswork,weinvestigatedtheportfolioselectionproblem withfourpracticalconstraintswhichlimitthenumberofassets ina portfolio,restricttheminimum andmaximum proportions ofassetsheldintheportfolio,requiresomespecificassetstobe includedintheportfolioandrequiretoinvesttheassetsinunitsof acertainsizerespectively.
Wehavedemonstratedthatmaintainingasecondary popula-tionofsolutionsetincombinationwithlearning-guidedcandidate solutiongenerationschemecontributetobetterperformanceover four existing well-known MOEAs, NSGA-II, SPEA2, PEAS-II and PAES.Theexperimentalresultsnotonlyshowthatthequalityof thegeneratedParetosetapproximationssignificantlyimproved, butalsothattheoverallcomputationtimecanbereduced.Asto theParetosetapproximation,theproposedsolutiongeneration schemeembeddinglearningmechanism,problemspecific heuris-ticsanddirection-basedsearchmethodsplaysamajorrole,while theefficiencyismainlybecausetheproposedalgorithmis compu-tationallycheapasitonlyusesasingleupdateateachgeneration. Performancewise,theproposedMODEwAwLalgorithmisnotonly capabletodeliverhigh-qualityportfoliosenrichedwithadditional constraintsbutalsoabletoefficientlysolvea reasonablesizeof assetupto1318.Theproposedalgorithmcouldbeappliedtoother practicalapplicationssuchasknapsackproblemwithrelevant con-straints.Forfuturework,theproposedalgorithmcanbeextended toincludeconstraintssuchastransactioncostandshortselling. Acknowledgements
ThisresearchwassupportedbytheSchoolofComputerScience, TheUniversityofNottingham.Theauthorswouldalsoliketothank anonymousreviewersfortheirhelpfulcomments.
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