Parallel Search for Combinatorial Optimization: Genetic Algorithms, Simulated Annealing, Tabu ...

14 Pages·2012·0.15 MB·English

Parallel Search for Combinatorial Optimization:
Genetic Algorithms, Simulated Annealing, Tabu Search
?
and GRASP
PanosM.Pardalos1,L.Pitsoulis1,T.Mavridou1,andMauricioG.C.Resende2
1 CenterforAppliedOptimizationandDepartmentofIndustrialandSystemsEngineering,
UniversityofFlorida,Gainesville,FL32611-6595USA
2 AT&TBellLaboratories,Room2D-152,600MountainAvenue,MurrayHill,NJ07974-2070
USA
Abstract. Inthispaper,wereviewparallelsearchtechniquesforapproximating
theglobaloptimalsolutionofcombinatorialoptimizationproblems.Recentde-
velopmentsonparallelimplementationofgeneticalgorithms,simulatedanneal-
ing, tabu search, and greedy randomized adaptive search procedures (GRASP)
arediscussed.
Keywords:ParallelSearch,Heuristics,GeneticAlgorithms,SimulatedAnneal-
ing,TabuSearch,GRASP,ParallelComputing.
1 Introduction
Searchtechniquesarefundamentalproblem-solvingmethodsincomputerscienceand
operationsresearch.Searchalgorithmshavebeenusedtosolvemanyclassesofprob-
lems, including path-(cid:2)nding problems, two-player games and constraint satisfaction
problems.Classicalexamplesofpath-(cid:2)ndingproblemsincludemanycombinatorialop-
timizationproblems(e.g.integerprogramming)andpuzzles(e.g.Rubic’scube,Eight
Puzzle). Chess, backgammon, and othello belong to the class of two player games,
while aclassic exampleofaconstraintsatisfactionproblemis the eight-queensprob-
lem.
ForNP-hardcombinatorialoptimizationproblems,exactsearchalgorithms,such
asbranchandbound,maydegeneratetocompleteenumeration.Forthatreason,exact
approacheslimitsustosolveonlymoderatelysized probleminstances,duetotheex-
ponential increase in CPU time when problem size increases. Therefore, in practice,
heuristic3 searchalgorithmsarenecessaryto(cid:2)ndsub-optimalsolutionstotheseprob-
lems. For large-scale problems, one of the main limitations of heuristic search is its
computationalcomplexity.Ef(cid:2)cient parallel implementationofsearch algorithmscan
signi(cid:2)cantlyincreasethesizeoftheproblemsthatcanbesolved.Whilethereisalarge
?invitedpaper,Workshopon(cid:147)ParallelAlgorithmsforIrregularlyStructuredProblems(cid:148)Lyon,
France(September4(cid:150)6,1995)
3Etymologically the word heuristic comes from the Greek word heuriskein to discover; the
Greek mathematician and inventor Archimedes (287(cid:150)212 B.C.) is known for the famous
Heureka!whenhediscoveredamethodtoverifythepurityofgold.
body of work on search algorithms, work on parallel search algorithms is relatively
sparse.
Inthispaper,weexploredifferentapproachesofparallelheuristicsearchforsolving
combinatorialoptimizationproblems.Wefocusonissuesofparallelizinggeneticalgo-
rithms,simulatedannealing,tabu(ortaboo)search,andGRASP(Greedyrandomized
adaptivesearchprocedures).Theseheuristicmethodshavebeenusedtoapproximately
solveawidespectrumofcombinatorialoptimizationproblems.
2 ParallelGenetic Algorithms
Inthe1960’s,biologistsbegantousedigitalcomputerstoperformsimulationsofge-
neticsystems.Althoughthesestudieswereaimedatunderstandingnaturalphenomena,
some were not too distant from the modern notion of a genetic algorithm (GA). Ge-
netic algorithms, as they are used today, were (cid:2)rst introducedby John Holland [23].
Geneticalgorithmstrytoimitatethedevelopmentofnewandbetterpopulationsamong
different species during evolution, just as their early biological predecessors. Unlike
moststandardheuristicalgorithms,GAsuseinformationofapopulationofindividuals
(solutions)whentheyconducttheirsearchforbettersolutionsandnotonlyinformation
fromasingleindividual.GAshavebeenappliedtoanumberofproblemsincombina-
torialoptimization.Inparticular,thedevelopmentofparallelcomputershasmadethis
aninterestingapproach.
A GA aims at computing sub-optimal solutions by letting a set of random solu-
tionsundergoasequenceofunaryandbinarytransformationsgovernedbyaselection
schemebiasedtowardshigh-qualitysolutions.Solutionstooptimizationproblemscan
oftenbecodedtostringsof(cid:2)nitelength.TheGAsworkonthesestrings[23].Theen-
codingisdonethroughthestructurenamedchromosomes,whereeachchromosomeis
madeupofunitscalledgenes.Thevaluesofeachgenearebinary,andaresometimes
calledalleles.Theproblemisencodedbyrepresentingallitsvariablesinbinaryform
and placing them together in a single chromosome. A (cid:2)tness function evaluates the
qualityofasolutionrepresentedbyachromosome.
There are several critical parts which strongly affect the success of genetic algo-
rithms:
(cid:150) Representationofthesolutions.
(cid:150) Generationoftheinitialpopulation.
(cid:150) Selectionofwhichindividualsinanoldpopulationthatwillbeallowedtoaffectthe
individualsofanewpopulation.Intermsofevolution,thisrelatestotheselection
ofsuitableparentsinthenewpopulation.
(cid:150) Geneticoperators,suchascrossoverandmutation.Thatis,howtorecombinethe
geneticheritagefromtheparentsinthepreviousgeneration.
IfP(t)denotesthepopulationattimet,theGAcanbedescribedasinFigure(1).
P(0)isusuallygeneratedatrandom.TheevaluationofapopulationP(t)involves
computingthe(cid:2)tnessoftheindividualsandcheckingifthecurrentpopulationsatis(cid:2)es
certainterminationconditions.Typesofterminationrulesinclude:
LayoutofGeneticAlgorithm
Input: Aprobleminstance
Output: A(sub-optimal)solution
1.t=0,initializeP(t),andevaluatethe(cid:2)tnessoftheindividualsinP(t)
2.while (terminationconditionisnotsatis(cid:2)ed)do
(a)t=t+1
(b)SelectP(t),recombineP(t)andevaluateP(t)
3.Outputthebestsolutionamongallthepopulationas
the(sub-optimal)solution.
Fig.1.LayoutofGeneticAlgorithm
(cid:150) Agiventimelimitwhichisexceeded.
(cid:150) Thepopulationisdominatedbyafewindividuals.
(cid:150) Thebestobjectivefunctionvalueofthepopulationsisconstantoveragivennumber
ofgenerations.
Duetotheirinherentparallelproperties,GAshavebeensuccessfullyimplemented
onparallelcomputers,introducingthiswayanewgroupofGAs,ParallelGeneticAl-
gorithms (PGAs). In a PGA the populationis divided into subpopulationsand an in-
dependent GA is performed on each of these subpopulations. Furthermore, the best
individualsofalocalpopulationaretransferredtotheothersubpopulations.Communi-
cationamongthesubpopulationsisestablishedtofacilitatetheoperationsofselection
andrecombination.Therearetwo typesofcommunication[34]: (1)amongallnodes
wherethebeststringineachsubpopulationisbroadcastedtoalltheothersubpopula-
tions,and(2)amongtheneighboringnodes,i.e.onlytheneighboringsubpopulations
receivethebeststrings.
ThemostimportantaspectsofPGAs,whichresultinaconsiderablespeeduprelative
tosequentialGAs,arethefollowing[25]:
(cid:150) Localselection,i.e.aselectionofanindividualinaneighborhoodisintroduced,in
contrastwiththeselectioninoriginalGAswhichisperformedbyconsideringthe
wholepopulation.
(cid:150) Asynchronous behavior which allows the evolution of different population struc-
turesatdifferentspeeds,possiblyresultinginanoverallimprovementofthealgo-
rithmintermsofCPUtime.
(cid:150) Reliabilityincomputationperformance,i.e.theperformanceofoneprocessordoes
notaffecttheperformanceoftheotherprocessors.
Jogetal.[25]considertwobasiccategoriesofparallelgeneticalgorithms:
1. Thecoarse-grainedPGAs,wheresubpopulationsareallocatedtoeachprocessorof
theparallelmachine.Theselectionandrecombinationstepsareperformedwithin
asubpopulation.
2. The(cid:2)ne-grainedPGAs,whereasingleindividualorasmallnumberofindividuals
areallocatedtoeachprocessor.
In the same referencea reviewof parallel geneticalgorithms appliedto the traveling
salesmanproblem(TSP) is presented.A PGA developedby SuhandGucht[51], has
beenappliedtoTSPsofgrowingsize(100-1000cities)problems.PGAswithoutselec-
tionandcrossover,socalledindependentstrategieswereused.Thesealgorithmsconsist
ofruningan(cid:147)unlimited(cid:148)numberofindependentsequentiallocal searchesinparallel.
PGAswithlowamountoflocalimprovementwereusedandperformedbetterinterms
ofqualitysolutionthantheindependentstrategies.Intermsofcomputationaltime,the
PGAshowednearlyalinear-speedupforvariousTSPs,usingupto90processors.The
algorithmswererunonaBBNButter(cid:3)y.
AnotherimplementationofaPGAappliedtotheTSPcanbefoundin[21],where
anasynchronousPGA,calledASPARAGOS,hasispresentedindetail.Anapplication
ofASPARAGOSwasalsopresentedbyMuhlenbein[33]forthequadraticassignment
problem(QAP)usingapolysexualvotingrecombinationoperator.ThePGAwasimple-
mentedonQAPsofsize30and36andforTSPsofsize40and50withknownsolutions.
ThealgorithmfoundanewoptimumfortheSteinberg’sproblem(QAPofsize36).The
numbersofprocessorsusedtorunthisproblemwere16,32,and64.The64processor
implementation(onasystem withdistributedmemory)gavebyfarthe best resultsin
termsofcomputationaltime.
Furthermore,BattitiandTecchiolli[5]presentedparallelizationschemesofgenetic
algorithmsand tabusearch forcombinatorialoptimizationproblemsand in particular
forquadraticassignmentandN-kproblemsgivingindicativeexperimentalresults.
Tanese [53] presents a parallel genetic algorithm implemented on a 64-processor
NCUBE/sixhypercube.Eachprocessorrunsthealgorithmoneachownsubpopulation
(coarse-grainedPGA),andsendsinanadjustablefrequencyaportionofgoodindivid-
ualstooneofitsneighboringprocessors.Eachexchangetakesplacealongadifferent
dimensionofthehypercube.ThePGAisappliedtoafunctionoptimizationproblemand
itsperformanceiscomparedwiththecorrespondingGA,whichisfoundtobesimilar.
Inaddition,thePGAachievescomparableresultswithnearlinear-speedup.
Petteyetal.[43]proposedan(cid:147)islandmodel(cid:148)thatrestrictsthecommunicationamong
thesubpopulationstosomeadjacentinterconnectedsubpopulations[30].ThePGAwas
tested on four of DeJong’s testbed of functions [14]. Population sizes of 50 to 800
were used, with the best answer in terms of quality solution, corresponding to size
50,whichisapproximatelythetheoreticaloptimalpopulationsize.Thealgorithmwas
implementedon anIntel iPSC,a message-basedmultiprocessorsystem with a binary
n-cubeinterconnectionnetwork.
Morerecently,LeeandKim[30]developedPGAs,basedontheislandmodel,for
solvingjobschedulingproblemswithgenerallyweightedearlinessandtardinesspenal-
ties(GWET),satisfyingspeci(cid:2)cproperties,suchastheV-shapedschedulearoundthe
duedate.Abinaryrepresentationschemeisusedtocodethejobschedulesintochro-
mosomes.A GAis developedbyparallelizingthe populationintosubgroups,each of
whichkeepsitsdistinctfeasibleschedules.Theinitialpopulationisconstructedsothat
the resulting genotypesequencesatis(cid:2)es the V-shaped schedule.Two differentrepro-
duction methods are employed, the roulette wheel selection and the N-best selection
method.Also,thecrossoverandmutationoperatorsareimplementedinparallel.Sev-
eral instances of problems with subpopulations of 30 chromosomes (jobs) and total
populationsizeof100to200whereusedtoevaluatetheef(cid:2)ciencyofthePGA.Theau-
thorsreportthattheroulettewheelselectionschemeperformsfarbetterthantheN-best
selectionintermsofqualitysolution,thoughtheN-bestselectiongivesgoodresultsin
terms of CPU time. The mutation operator seems to improve the performanceof the
PGA.Thepaperconcludes,showingthesuperiorperformanceoftheparallelalgorithm
whencomparedtothesequentialGA.IntermsofCPUtime,theparallelizationofthe
populationintoseveralgroupsspeedsuptheconvergenceoftheGAsasthesizeofthe
problemincreases.
Parallelgeneticalgorithmshavebeenappliedtothe graphpartitionproblem[29],
schedulingproblems[11],andglobaloptimizationproblems[49].
3 ParallelSimulatedAnnealing
Since the simulated annealing method (SA) was proposed several years ago by Kir-
patricketal.[27],basedonthepioneeringworkofMetropolisetal.[32],muchresearch
hasbeenaccomplishedregardingitsimplementation(sequentialandparallel)tosolvea
varietyofdif(cid:2)cultcombinatorialoptimizationproblems.Simulatedannealingisbased
ontheanalogybetweenstatisticalmechanicsandcombinatorialoptimization.Theterm
annealingrefers to the process of a thermal system by (cid:2)rst melting at high tempera-
tures andthenloweringthetemperatureslowly basedon an annealingschedule,until
the vicinity of the solidi(cid:2)cation temperature is reached, where the system is allowed
toreachthegroundstate(thelowestenergystateofthesystem).Simulatedannealing
is a simple Monte Carlo approachto simulate the behavior of this system to achieve
thermalequilibriumatagiventemperatureinagivenannealingschedule.Thisanalogy
has been applied in solving combinatorialoptimization problems.Given an objective
function f(x) over a feasible domain D, a generic simulated annealing algorithm for
(cid:2)ndingtheglobalminimumof f(x)isgiveninFigure2.
Anintroductiontogeneralconceptsofparallelsimulatedannealingtechniquescan
be found in Aarts and Korst[1]. Several parallel algorithms based on SA have been
implemented for a variety of combinatorial optimization problems. In the context of
annealing,aparallelimplementationcanbepresentedintwoforms[47]:(1)functional
parallelism, which uses multiple processors to evaluate different phases of a single
move, (2) data parallelism, which uses different processors or group of processors
to proposeand evaluate movesindependently.The second formhas the advantageof
(cid:147)easilyscalingthealgorithmtolargeensemblesofprocessors.(cid:148)
Thestandard-cellapproachisasemi-customdesigningmethodinwhichfunctional
buildingblockscalledcellsareusedtoconstructapartof,oranentire,VLSIchip[47].
Twobasicapproacheshavebeenappliedtotheplacementproblem[47](cid:150)constructive
LayoutofSimulatedAnnealing
Input: Aprobleminstance
Output: A(sub-optimal)solution
1.GenerateaninitialsolutionatrandomandinitializethetemperatureT.
2.while (T >0)do
(a)while (thermalequilibiumnotreached)do
(i)Generateaneighborstateatrandomandevaluatethechangein
energylevelDE.
(ii)IfDE<0updatecurrentstatewithnewstate.
(iii)IfDE 0updatecurrentstatewithnewstate
(cid:21)
DE
withprobabilityeK(cid:0)BT.
(b)DecreasetemperatureT accordingtoannealingschedule.
3.Outputthesolutionhavingthelowestenergy.
Fig.2.LayoutofSimulatedAnnealing
methodsanditerativemethods.TheSAtechniquebelongstothesecondgroup,asan
approachthatusesprobabilistichillclimbingtoavoidlocalminima.TheTimberWolf
programhasbeenproposedasaversionofSAimplementedforthecellplacementprob-
lem.Ithasbeenshowntoprovideenormouschipareasavingscomparedtothealready
existedstandardmethodsofcelllayout.Jones andBanerjec[26] developeda parallel
algorithmbasedonTimberWolf,wheremultiplecellmovesareproposedandevaluated
byusingpairsofprocessors.ThealgorithmwasimplementedonaiPSC/1Hypercube.
Roseetal.[45]presentedtwoparallelalgorithmsforthesameproblem.TheHeuristic
Spanningapproachreplaces the high temperatureportionof simulated annealing,as-
signingcellsto(cid:2)xedsub-areasofthechip.TheSectionAnnealingapproachisusedto
speed-upthelowtemperatureportionofsimulatedannealing.Theplacementhereisge-
ographicallydividedandthepiecesareassignedtoseparateprocessors.Otherparallel
algorithmsbasedonSAandimplementedforthecellplacementproblemcanbefound
in[8,47,55].
Greening [22] examines the asynchronousparallel SA algorithm in relation with
the effects of calculation errors resulting from the parallel implementation or an ap-
proximate cost function. More analytically, the relative work analyzes the effects of
instantaneousandaccumulatederrors.The(cid:2)rstcategorycontainstheerrorswhichre-
sultasthedifferencebetweenthetrueandinaccuratecostscomputedatagiventime.An
accumulatederroristhesumofastreamofinstantaneouserrors.Theauthorprovesa
directconnectionbetweentheaccumulatederrorsmeasuredinpreviousresearchwork,
andannealingproperties.
Mostrecently,BoissinandLutton[6]proposedanewSAalgorithmthatcanbeef(cid:2)-
cientlyimplementedonamassivelyparallelcomputer.Acomparisontothesequential
algorithmhasbeenpresentedbytestingbothalgorithmsontwoclassicalcombinatorial
optimization problems: (1) the quadratic sum assignment problem and (2) the mini-
mizationofanunconstrained0 1quadraticfunction.Thenumericalresults showed
(cid:0)
thattheparallelalgorithmconvergestohigh-qualitysuboptimalsolutions.Fortheprob-
lemofminimizationofquadraticfunctionswith1000variables,theparallelimplemen-
tationona16KConnectionMachinewas3.3timesfasterthanthesequentialalgorithm
implementedonaSPARC2workstation,forthesamequalityofsolution.
4 ParallelTabuSearch
Tabu search (TS), (cid:2)rst introducedby Glover[18, 19], is a heuristicprocedureto (cid:2)nd
goodsolutionstocombinatorialoptimizationproblems.Atabulistisasetofsolutions
determinedbyhistoricalinformationfromthelastt iterationsofthealgorithm,wheret
is(cid:2)xedorisavariablethatdependsonthestateofthesearch,oraparticularproblem.
Ateachiteration,giventhecurrentsolutionxanditscorrespondingneighborhoodN(x),
theproceduremovestothesolutionintheneighborhoodN(x)thatmostimprovesthe
objectivefunction.However,movesthatleadtosolutionsonthetabulistareforbidden,
oraretabu.Iftherearenoimprovingmoves,TSchoosesthemovewhichleastchanges
theobjectivefunctionvalue.Thetabulistavoidsreturningtothelocaloptimumfrom
whichtheprocedurehasrecentlyescaped.Abasicelementoftabusearchistheaspi-
rationcriterion,whichdetermineswhenamoveisadmissibledespitebeingonthetabu
list. One terminationcriterionforthe tabu procedureis a limit in the numberof con-
secutivemovesforwhichnoimprovementoccurs.Amoredetaileddescriptionofthe
mainfeatures,aspects,applicationsandextensionsoftabusearch,canbefoundin[20].
Several implementationsof parallel algorithmsbased on TS have been developedfor
classicaloptimizationproblems,suchasTSPandQAP,whichwillbediscussedbelow.
Given an objectivefunction f(x) overa feasible domainD, a generictabu searchfor
(cid:2)ndinganapproximationoftheglobalminimumof f(x)isgiveninFigure3.
Taillard[52]presentstwoimplementationsofparallelTSforthequadraticassign-
ment problem. Computationalresults on instances of size up to 64 are reported.The
formofTSconsideredisclaimedtobemorerobustthanearlierimplementations,since
it requires less computational effort and uses only the basic elements of TS (moves
toneighboringsolutions,tabulist,aspirationfunction).Theneighborhoodusedisa2-
exchangeneigborhood.Thetabulistismadeupofpairs(i;j)ofinterchangedunitsboth
ofwhichareplacedatlocationstheyhadoccupiedwithinthelastsiterations,wheres
isthesizeoftabulist.Thesizeschangesitsvaluerandomlyinanintervalduringthe
search.Theaspirationcriterionisintroducedtoallowthetabumovestobechosen,if
both interchangedunits are assigned to locations theyhave not occupiedwithin thet
mostrecentiterations.Inthe(cid:2)rstmethod,theneighborhoodisdividedinto ppartsof
LayoutofTabuSearch
Input: Aprobleminstance
Output: A(sub-optimal)solution
1.Initialization:
(a)Generateaninitialsolutionxandsetx(cid:3)=x;
(b)InitializethetabulistT=0/;
(c)Setiterationcountersk=0andl=0;
2.while (N(x) T =0/)do
n 6
(a)k=k+1;l=l+1;
(b)SelectxasthebestsolutionfromsetN(x) T;
n
(c)If f(x)< f(x(cid:3))thenupdatex(cid:3)=xandsetl=0;
(d)Ifk=korifl=lgotostep3;
3.Outputthebestsolutionfoundx(cid:3);
Fig.3.LayoutofTabuSearch
approximatelythesamesize.Eachpartisdistributedforevaluationtooneof pdiffer-
ent processors.Using a numberof processorsproportionalto the size of the problem
(precisely p=n=10),thecomplexityisreducedbyafactorofn,wherenisthesizeof
theproblem.Moreover,the computationalresults showedimprovementto the quality
ofsolutionformanylargeproblemsandthebestpublishedsolutionsofotherproblems
have been found. The second method performs independentsearches, each of which
startswithadifferentinitialsolution.Theparallelalgorithmwasimplementedonaring
of 10 transputers (T800C-G20S). The computational complexity of this algorithm is
lessthanthatofearlierTSimplementationsforQAPbyafactorn.
Anotherparalleltabusearchalgorithmforsolvingthe quadraticassignmentprob-
lemhasbeendevelopedbyChakrapaniandSkorin-Kapov[9].Thealgorithmincludes
elementsofTS,suchas aspirationcriterion,dynamicallychangingtabulist sizes and
long-termmemory[18].Anewintensi(cid:2)cationstrategyisproposed,basedontheinter-
mediateterm memory,restricting the searchesin a neighborhood.This results in less
amountof computationaleffortduring one iteration since the proceduredoes not ex-
aminetheentireneighborhood.Amassivelyparallelimplementationwastestedonthe
ConnectionMachineCM-2forlargeQAPs ofsizerangingfromn=42to100,using
n2processors.Thenewtabustrategygavegoodresultsintermsofqualityofsolutions.
Forproblemsuptosize90,itobtainedthebestknownsolutionsorclosetothose.For
sizen=100,everypreviouslybestsolutionfound,wasimprovedupon.
Battiti and Tecchiolli [5] describe a parallelization scheme for tabu search called
reactive tabu scheme, and apply it to the quadratic assignment problem and the N-
k problem with the objective of achieving a speedup in the order of the number of
processors.Inthereactivetabusearch,eachprocessorexecutesandindependentsearch.
Furthermore,thesameproblemsaresolvedbyaparallelgeneticalgorithminwhichthe
interactionbetweendifferentsearchprocessesisstrongbecausethegenerationofnew
candidatepointsdependsontheconsiderationofmanymembersofthepopulation.
A tabu search for the traveling salesman problem has been proposed by Fiechter
[17],andwastestedoninstanceshaving500to10000vertices.Anewestimationofthe
asymptoticnormalizedlengthoftheshortesttourthroughpointsuniformlydistributed
intheunitsquareisgiven.Numericalresultsandspeedupsobtainedbytheimplemen-
tationofthealgorithmona networkoftransputersshowthe ef(cid:2)ciencyofthe parallel
algorithm.
5 ParallelGRASP
A GRASP[16] is an iterative process for (cid:2)nding approximate solutions to combina-
torialoptimization.Let,asbefore, f(x)denotetheobjectivefunctiontobeminimized
overafeasibledomainD.AgenericlayoutofGRASPisgiveninFigure4.TheGRASP
LayoutofGRASP
Input: Aprobleminstance
Output: A(sub-optimal)solution
1.Initialization:setx(cid:3)=¥;
2.while (stoppingcriterionnotsatis(cid:2)ed)do
(a)Constructagreedyrandomizedsolutionx;
(b)Findlocalminimumx(cid:152)inneighborhoodN(x)ofx;
(c)If f(x(cid:152))< f(x(cid:3))thenupdatex(cid:3)=x(cid:152);
3.Outputthebestsolutionfoundx(cid:3);
Fig.4.LayoutofGRASP
iterationsterminatewhensomestoppingcriterion,suchasmaximumnumberofitera-
tions, is satis(cid:2)ed. Each iteration consists of a construction phase, and a local search
phase. If an improvedsolution is found,then the incumbentis updated.A high-level
descriptionofthesetwophasesisgivennext.
Intheconstructionphase,afeasiblesolutionisbuiltup,oneelementatatime.The
choice of the next element to be added is determined by ordering all elements in a
candidatelist withrespecttoagreedyfunction.An elementfromthecandidatelistis
randomlychosenfromamongthebestcandidatesinthelist,butisnotnecessarilythe
topcandidate.Figure5displaysagenericlayoutfortheconstructionphaseofaGRASP.
ThesolutionsetSisinitializedempty,andtheconstructioniterationsarerepeateduntil
LayoutofGRASPConstructionPhase
Input: Aprobleminstanceandpseudorandomnumberstream
Output: A(sub-optimal)solution
1.Initialization:setsolutionS=0/;
2.while (solutionconstructionnotdone)do
(a)Usinggreedyfunction,makerestrictedcandidatelist(RCL);
(b)Atrandom,selectelementsfromRCL;
(c)Placesinsolution,i.e.S=S s ;
[f g
(d)ChangegreedyfunctiontotakeintoaccountupdatedS;
3.OutputthesolutionxcorrespondingtosetS;
Fig.5.LayoutofGRASPConstructionPhase
the solution is built. To do this, the restricted candidate list is setup. This candidate
list contains a subset of candidates that are well ordered with respect to the greedy
function. A candidate selected from the list at random is added to the solution. The
greedyfunctionisadaptedtotakeintoaccounttheselectedelement.
Thesolutionsgeneratedintheconstructionarenotguaranteedtobelocallyoptimal
and therefore local search is applied to produce a locally optimal solution. Figure6
illustratesagenericlocalsearchprocedure.
GRASP can be easily implemented on an MIMD multi-processor environment.
Eachprocessorcanbeinitializedwithitsowncopyoftheprocedure,theproblemdata,
and independent pseudo-random number sequences. The GRASP iterations are then
independently executed in parallel and the best solution found over all processors is
theGRASP solution.Thisapproachwas implementedbyPardalos,Pitsoulis,andRe-
sende[39]to approximatelysolve instances ofthe QAP ofdimensionup to n=128.
OnaKendallSquareKRS-1parallelcomputerwith128processors(ofwhichonly64
wereutilized),theauthorswereabletoachieveonaverageaspeedupfactorof62and
speedupfactorsof130and221onsingleinstancesofdimensionsn=30andn=22,
respectively.
Another approachto implement GRASP parallel was proposed by Feo, Resende,
andSmith[15],wheretheproblembeingoptimizedisdecomposedintomanysmaller
problems, and each processor is given a set of small problems, that are solved with