the convex hull relaxation for nonlinear integer … integer minimization ... while solving a linear...

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TheConvexHullRelaxationforNonlinearIntegerPrograms

WithLinearConstraints

By

AykutAhlatciogluand

MoniqueGuignard1

OPIMDepartmentTheWhartonSchool

UniversityofPennsylvania

Draft20September2007

Donotquotewithouttheauthors’permission

1ResearchpartiallysupportedunderNSFGrantDMI‐0400155.

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1. INTRODUCTIONInthispaperweintroducearelaxationmethodforcomputingbothalowerboundonthe

optimalvalueofanonlinearintegerminimizationprogram(NLIP),andgoodintegerfeasiblesolutions.Foralinearintegerprogram(LIP),anoptimalintegersolutionisalsooptimalovertheconvexhullofallintegerfeasiblesolutions,butthisisnotusuallythecaseforNLIPs.Rather,theminimizationoverthisconvexhullyieldsarelaxationoftheNLIP,whichwewillcalltheConvexHull(CH)Relaxation.WhilewedefinethisrelaxationforarbitraryNLIPs,forcomputationalreasons,werestrictourattentiontoconvexminimizationproblemswithlinearconstraints,andweshowthatthelowerboundcanthenbecomputedusinganyversionofsimplicialdecomposition,withsub‐problemsthathavethesameconstraintsastheNLIP,butwithlinearobjectivefunctions.Iftheseareeasiertosolvethantheirnonlinearcounterpart,aswouldbethecaseforinstancefornonlinear0‐1knapsackproblems,theboundmaybetightandrelativelyinexpensivetocompute.

Abyproductofthisprocedureisthegenerationoffeasibleintegerpoints,whichprovideatight

upperboundtotheoptimalvalueoftheproblem.Indeedsincenoconstraintsarerelaxed,anyintegersolutiongeneratedwhilesolvingalinearintegerproblemisalsoanintegerfeasiblesolutiontotheoriginalnonlinearproblem.Thus,unlikeLagrangeanorprimalrelaxations,CHRprovidesintegerfeasiblesolutionsatnoextracost.Fromournumericalexperiments,thesesolutionstendtobeofexcellentquality.

WhatmakesthisrelaxationveryspecialisthatcontrarytoLagrangeanrelaxation(HeldandKarp,

1970,1971)ortoprimalrelaxation(Guignard,1994,2003(p.183‐185)),ofwhichitisanextremecase,thisrelaxationdoesnotdualizeortreatseparatelyanyconstraints.Whilefornonlinearintegerproblems,itcannotbedirectlycomparedwithLagrangeanrelaxation,italwaysprovidesaboundatleastasgoodasanyprimalrelaxation.However,if(1)thelinearsubproblemsaredifficulttosolve,(2)theobjectivefunctionisnonconvex,and/or(3)therearenonlinearconstraints,thenoneshouldconsiderusingaprimalrelaxationinstead.Inothercases,thisnewapproachappearsveryattractive.

Inthepaper,wefirstdefinetheCHrelaxation(CHR)insection2,analyzetheapplicationof

simplicialdecompositiontotheCHRprobleminsection3,givedetailsonthealgorithminsection4,discussthecomputationofupperandlowerboundsontheoptimalvalueinsection5,anddescribetheapplicationoftheapproachtoconvexquadraticknapsackproblemsinsection6.Theremainingsectionspresentthecomputationalresults.

2. PRELIMINARIESANDNOTATION

Theproblemwhichwillbeinvestigatedwillbethefollowingnonlinearintegerprogram

(NLIP)

where

3

isanonlinearconvexfunctionofx,avectorpfR^n,

Aisan constraintmatrix,

isaresourcevectorin ,

isasubsetofR^nspecifyingintegralityrestrictionson .

Definition1

WedefinetheConvexHullRelaxationof(NLIP)tobe

. .

Theproblem(CHR)isnotingeneralequivalentto(NLIP)when isnonlinear,becausean

optimalsolutionof(CHR)maynotbeinteger,andthereforenotfeasiblefor(NLIP).However,itiseasytoseethat(CHR)isindeedarelaxationto(NLIP),as

(1){ | } { | }x Y Ax b Co x Y Ax b∈ ≤ ⊆ ∈ ≤

(11)

Thisrelaxationisaprimalrelaxation,inthex‐space,anditisanextremecaseoftheprimalrelaxationfornonlinearintegerproblemsintroducedinGuignard(1994).Itisactuallyaprimal

relaxationthatdoesnot“relax”anyconstraint.Thedifficultyinsolving(CHR)comesfromtheimplicitformulationoftheconvexhull.

Howevertheideaofdecomposingtheproblemintoasub‐problemandamaster‐problem,first

introducedbyFrank&Wolfe(1956),andfurtheredbyVonHohenbalkenwithSimplicialDecomposition(1973),andHearnetal.withRestrictedSimplicialDecomposition(1987),providesanefficientwayofsolving(CHR)tooptimality,bysolvingasequenceoflinearintegerproblemsandof

nonlinearproblemsoverasimplex.Primalrelaxationsthatalsorelaxconstraintsrequireamorecomplicatedscheme,suchasthatdescribedinContesseandGuignard(1995,2007)andAhn,ContesseandGuignard(1998,2006),whichuseanaugmentedLagrangeanapproach,withsimplicial

decompositionusedateachiteration.Bycontrast,here,duetotheabsenceofrelaxedconstraints,onlyonecalltosimplicialdecompositionisneeded.

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3. APPLYINGSIMPLICIALDECOMPOSITIONTOTHECHRPROBLEM

3.1 Assumptions

Thereareseveralassumptionsthatmustbeimposedonthe(NIP)formulationinorderforthesimplicialdecompositiontechniquetoeffectivelysolve(CHR)tooptimality.Theseare:

(i)Compactnessandconvexityofthefeasibleregion

Compactnessandconvexityofthefeasibleregionallowswritinganyelementinthefeasiblespaceasaconvexcombinationofitsextremepoints.However,itisnothardtorelaxthe

boundedassumptionusingextremerays2.

(ii)ConvexityoftheobjectivefunctionSimplicialdecompositionguaranteesconvergencetoalocalminimum.However,alocal

minimumisnotnecessarilyalowerboundfortheoptimalvalueof(NLIP).Convergenceoftheglobalminimumisensuredifthefunctionistakentobepseudo‐convex.3Inthispaper,wewillassumethatobjectivefunctionsareconvex,ormadeconvexviaconvexification.

(iii)Linearityoftheconstraintset.

Althoughtheobjectivefunctioncanbenonlinear,wecannotatthispointallownonlinearconstraints.

3.2 Subproblem

Thefirstpartofthedecompositionproblemisthesub‐problem,andcanalsobeviewedasa

feasibledescentdirectionfindingproblem.4Assumeweareatafeasiblepointof(CHR),callit .

Forthe iterationofsimplicialdecomposition,wemustfindafeasibledescentdirection

for ,apolyhedron,bysolvingthefollowingproblem

WewillcallthistheConvexHullSub‐problem(CHS).NotethatCHSisalinearprogram.

Therefore,unlikeinnonlinearprogramming,(CHS)hasanequivalentintegerprogram,whichwewillcalltheIntegerProgrammmingSubproblem(IPS)

2See,forinstance,Section6.3ofHearnetal.(1987)3Forfurtherdiscussiononpseudo‐convexitysee,forinstance,Bazaraaetal.(1979)p.106andseq.4Forfurtherdiscussiononfeasibledirectionssee,forinstance,Bertsekas(2003)p.214andseq.

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Solving(IPS),formanytypesofintegerprogrammingproblemsisconsiderablyeasier

comparedwithsolving(NLIP).Thesolutionto(IPS)isanextremepointoftheconvexhull,unless

isoptimalfortheconvexhullrelaxation(CHR)problem.Therefore,ateachiterationweobtainafeasiblepointtotheoriginal(NLIP)problem.Convergencetotheoptimalsolutionwillbediscussedin

sectionIV.If isnotoptimal,weproceedtothemasterproblem.

3.3MasterProblem

Considerthefollowingnonlinearprogrammingproblemwithonesimpleconstraint,whichwecalltheMasterProblem(MP).

isthe matrixcomprisedofasubsetofextremepointsoftheconvexhull,alongwithoneof

thecurrentiterates orapastiterate.Thereare suchpointsin .Notethatinthehypothetical

casewhereweknowalltheextremepointsoftheconvexhull(MP)wouldhavebeenequivalentto(CHR).Naturally,ifthemethodrequiredsuchequivalence,therewouldbenopointinusingittosolve(CHR).Luckily,anypointwithinaconvexhullofasetcanbedescribedasaconvexcombinationofatmost pointswithinthatset,aresultofCaratheodoryTheorem5.Therefore,theoptimal

pointcanbewrittenasaconvexcombinationofasubsetofextremepoints.Simplicialdecompositiontakesadvantageofthisobservation,introducingonlyoneextremepointobtainedfromthesubproblemperiteration.Thenatthemasterproblemstage,(MP)issolved,whichisaminimizationproblemovera dimensionalsimplex.Iftheoptimalsolutionof(CHR)iswithinthissimplex,

thenthealgorithmterminates.Ifnot,theoptimalsolutionof(MP)willbethenextiterate,

whichcanbefoundusingthefollowingformula:

Thenwegobacktothesubproblem,findanotherextremepointandincreasethedimension

ofthesimplexfor(MP).ItmayseemasifaconsiderablenumberofextremepointshavetobeincludedinX,beforefindingtheoptimalsolution.Fortunately,thisisnotthecase,aswillbeshown

5ForfurtherdiscussiononCaratheodoryTheoremsee,forinstance,Bazaraa(1979)p.37andseq.

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atsection8.Thiscanbeperhapsexplainedbythewayextremepointsareintroducedtothe

matrix.Ateachsub‐problem,theextremepointchosen istheonewhichyieldsthesteepest

descentdirectionas isminimalamongallsuchdirections.Thereforeateach

iteration,wearequicklyprogressingtowardtheoptimalsolution,incontrastwhatwouldhappenifextremepointsweretobechosenarbitrarily.

Forsomepathologicalcases,puttingnorestrictiononrcouldpotentiallyposecomputational

problems.Restrictedsimplicialdecomposition,introducedbyHearnetal.(1987)putsarestrictiononthenumberofextremepointswhichcanbekept.However,evenforsuchpathologicalcases,therearecertaintrade‐offsbetweenrestrictedsimplicialdecompositionandunrestrictedsimplicial

decomposition.Discussingthesetradeoffsisbeyondthescopeofthispaper.

3.4ConvergencetotheOptimalSolutionofCHR

Becausetheobjectivefunctionisconvex,thenecessaryandsufficientoptimalitycondition

for tobetheglobalminimumis6,

Lemma2ofHearnet.al(1987)provesthatif isnotoptimal,then ,so

thatthesequence ismonotonicallydecreasing.FinallyLemma3ofHearnetal.(1987)shows

thatanyconvergentsubsequenceof willconvergetotheglobalminimum.Theresultisproved

usingcontradictionthatonecannothaveasubsequencesuchthatwhere .

4. ALGORITHM

Thealgorithmusedinthisstudyfollowstherestrictedsimplicialdecomposition(Hearnetal.

1987).TheparameterRdenotesthemaximumnumberofextremepointsallowedtobeusedinsolvingthemasterproblem.Inthetestrunsdoneforthispaper,thenumberofextremepointsstoredinthematrixXwasmanageable,sothatweputnolimitonR,makingthealgorithmbelow

equivalenttotheunrestrictedsimplicialdecompositionmethodofvonHohenbalken(1977)7.ThestoppingconditionforthealgorithmistakenfromContesse&Guignard(2007).

Notethatinthisnotation:

(1) isthecollectionofextremepointsstoredatiterationk,

(2) storesatmostonepoint.Itcouldbeempty,acurrentiterateorapast

iterate.

Then,withthisnotationthemasterproblemintroducedinsection3.3willbe:

6Forproofsee,forinstance,Bertsekas(2003)p.194

7Theoriginalnameofthemethodis‘SimplicialDecomposition’,butinthispaperIwillcallit‘UnrestrictedSimplicialDecomposition’todifferentiateitfromthe‘RestrictedSimplicialDecomposition’

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Animportantpointtonoteisthatwediscardthosepointswithin with

aftersolvingthemasterproblem.Thispreventsanexcessiveincreaseinthenumberof

extremepointsstoredin .

Step0:Takeafeasiblepoint .Set

Step1:Solve andlet

(i)

(ii)

Step2:Let .

Step3:If , isa

solution,thenterminate.Otherwise,gotostep1.

5. CALCULATINGLOWERANDUPPERBOUNDS

AsstatedinDefinition1,(CHR)isarelaxationtothe(NLIP).SimplicialDecompositionfindsanoptimalsolution,say, ,to(CHR),andthisprovidesalowerboundonv(NLIP):

Ontheotherhand,ateachiterationkofthesubproblemanextremepointoftheconvexhullisfound,whichisanintegerfeasiblepointof(NLIP).Eachpoint yieldsanUpperBound(UB)tothe

optimalvalueofthe(NLIP),andthebestupperboundonv(NLIP)canbecomputedas

6. APPLICATIONOFCHRMETHODTOCONVEXQUADRATICANTI‐KNAPSACKPROBLEMS

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AsanexampleproblemweimplementedCHRtofindalowerboundontheoptimalvalueof

quadraticanti‐knapsackproblems.TheQuadraticAnti‐KnapsackProblem(QKP)isasfollows:

whereisthelinearcostincurredbyselectingitem .

isthequadraticcostincurredbyselectingitems and concurrently.

isthespacefilledifitem isselected.

bisthetotalminimumspaceneededtobefilledintheknapsack.

Followingthediscussionabove,theConvexHullRelaxation(CHR),itssubproblem(IPS)anditsmasterproblem(MP)willbeasfollows:

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Wecancalculatethenextiterate,usingthefollowingformula:

7. GENERATIONOFTHEDATA

Asnotedpreviously,theobjectivefunctionneedstobetakenconvex,fortheCHRmethodtoproducealowerboundforthe(NLIP)problem.Tocomeupwithconvexobjectivefunctionsfor(QKP),weusedthefollowingfactwhichenablesustoproducepositivedefinitematrices.

FACT1

Assumewehavethefollowingmatricesathand:

(i)An diagonalmatrixDwithpositiveentries,hencepositivedefinite(ii)An matrixAwithrankm,where

Then ispositivedefinite.(Proofatsection10.2AppendixB)

Thenitisnothardtogeneratepositivedefinitematricesusingtherandomnumbergenerator

(RNG)installedinGAMSforthenonlinearpartoftheobjectivefunctionrepresentedby .Similarly,dataforthelinearpartoftheobjectivefunctionandtheconstraint

vectoraregeneratedthroughRNG.TheelementsofthematrixCispopulated

around , isuniformlydistributedon and isuniformly

distributedon .

Therunshavebeenmadefor100,200and400decisionvariables,whichareitemsinthisproblem.Foreachofthese,therighthandsidedenotedby ischangedtoobservehowthecapacity

oftheknapsackinfluencestheperformanceoftheCHRmethod.FivedifferentvaluesofbaretriedandrepresentedbycaseIthroughcaseV.CaseIrepresentstheknapsackwithminimumcapacity,andCaseVrepresentsthemaximum.Oneshouldnotethat ,thetotalspacefilledifallitems

weretakenintotheknapsack,increaseinproportiontothenumberofvariables.Thereforeforeachcasetorepresenttheequivalenteffect,valuesfor aresetinproportiontothenumberofitems.

ThesevaluescanbeseeninTable1below.8Table1‐bvaluesfordifferentcases

No.of

variables

CaseI CaseII CaseIII CaseIV CaseV

8Foramoredetailedaccountoftheresults,seeSection10.1AppendixA.

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100 5000 125,000 500,000 1,250,000 2,500,000

200 10000 250,000 1,000,000 2,500,000 5,000,000

400 20000 500,000 2,000,000 5,000,000 10,000,000

TheproblemisrunusingtheCHRmethod,aswellasbuilt‐inMINLPsolversofGAMS,namely

CPLEXandAlphaECP.Attheendcomparisonsaremadefortheirperformances.Resultsbeloware

obtainedusingGAMS2.25onDellOptiPlex745.

8. DISCUSSIONOFTHERESULTS

8.1 ImprovementOvertheContinuousBound Theimprovementoverthe

continuousboundprovidedbyCHRboundisveryhigh,whenintroductionofoneitemissufficienttosatisfytheknapsackconstraint.(CaseI).Thisisexpected,andthereforenot

takenintoconsiderationwhencalculatingthepercentageimprovementvaluesbelow.Table1belowshowspercentageimprovementsforallthe15instances.

Table1‐%Improvementoverthecontinuousbound


100 3310 13.6 12.1 24.5 53.2

200 1392 19.4 0.25 2.00 0.14

No.ofV

ariable

400 537 1.24 0.19 0.30 0.06

Onecouldtalkofdecreaseofpercentageimprovementasnumberofvariablesincrease.TheonlyexceptiontothisisatcaseII.Ontheotherhand,itishardtoextractasimilarstrongrelationbetweentheknapsackcapacityandpercentageimprovement.ExcludingcaseI,averagepercentage

improvementfortheremaining12runsis:

8.2 GapBetweentheCHRLowerBoundandtheOptimalValue

TheCHRboundprovidesaremarkablytightboundfortheoptimalvalue.Table2showsthatcalculatingtheCHRboundcouldindeedbeusefulinprovidingatighterlowerboundcomparedtotheoneobtainedthroughcontinuousrelaxation(ComparewithTable1).

Table2‐%GapbetweentheCHRlowerboundandtheoptimalvalue


No.of

Variable

100 9.41 4.94 0.70 0.12

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200 14.9 2.64 0.05 0.03

400 13.7 1.29 0.09

ExcludingcaseI,averagegapbetweenCHRboundandtheoptimalvalueis

Weseethataswerequiremorevariablestogetintotheknapsack(notethedecreasefromcaseItocaseV),thegapclosesup.

8.3 GapBetweentheCHRUpperBoundandtheOptimalValue

TheCHRupperboundalsoprovidesaverytightboundfortheoptimalvalue.Infact,formostoftheinstanceswetried,itreturnedtheoptimalsolution.Howevertoverifythisremarkable

strengthoftheupperbound,thereisneedfortestingthealgorithmforothertypeofproblems,especiallythoseknowntobedifficulttosolvetooptimality.

Table3‐%GapbetweentheCHRUpperBoundandtheoptimalvalue


100 0.0 0.0 0.0 0.0 0.0

200 0.0 0.12 0.0 0.0 0.0

No.ofV

ariable

400 0.0 0.0* 0.0 0.0 0.0

8.4 RunTimeandReliabilityoftheCHRMethod

Asimportantasobtainingtightbounds,isthecostofobtainingthatbound,aswellasthe

reliabilityoftheboundortheoptimalvalueobtained.TheobviousmeasureofcostistheCPUtimeittakesforthealgorithmtoconvergeandreturnupperandlowerbounds,andhowthiscompareswiththeexistingMINLPsolversinGAMS.Reliabilitycanbemeasuredwhetherthealgorithmisableto

turninaboundoranoptimalsolutionforallinstanceswithprecisioninareasonableamountoftime.Table3belowshowsCPUtimeinsecondsforCPLEX,AlphaEcpandCHR.Comparingtheaverageruntimes,onecouldseethatAlphaEcpisnotcomparabletoCPLEXorCHR.AlthoughCPLEX

performsbetterthanCHRonaverage,thisismostlybecauseofoneparticularinstancewhereCHRperformspoorly(400variable,caseIII).Excludingtheworstperformances,CHRindeedperformsbetterthanbothCPLEXandAlphaEcp.Infact7outofthe15runs,CHRterminateswiththeshortest

runtime.OneshouldalsonotethatCPLEXisonlyabletosolveproblemswithquadraticobjective

functions,whereasCHRdoesnottakeadvantageofthequadracityoftheobjectivefunction,andwillworkjustaswellforanon‐quadraticobjectivefunction.Table3alsoshedslightonthereliabilityofthesealgorithms.Fortwoinstances,AlphaEcpfailstoterminateinlessthan30minutes.Ontheother

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hand,CPLEXterminatedwithafeasiblebutnotoptimalintegersolutionforfourinstancesdueto

worseningoftheobjectivefunctionofthenonlinearobjectivefunction.

Table3‐Runtimesinseconds

#ofVariables Case CPLEX AlphaEcp CHR100 I 1.01 10 1.07

100 II 1.10 9 0.62100 III 1.26* 251 2.44

100 IV 0.76 32 2.84

100 V 0.83 2 0.57200 I 2.40 53 2.95

200 II 4.10* 1611 4.28200 III 2.98 2 1.4

200 IV 2.59 25 6.81200 V 2.40 3 1.78

400 I 15.16 376 5.65

400 II 19.54* N/A** 27.79400 III 15.77* N/A** 90.68

400 IV 10.25 4 1.00400 V 11.04 7 8.12

Average(all) 6.08 183.46 10.53Average(excludingthe

worst)5.12 64.5 4.8

*CPLEXWarning:ThesearchwasstoppedbecausetheobjectivefunctionoftheNLPsubproblemstartedtodeteriorate**Unabletoterminateinlessthan30minutes

Especially,intwoinstancesCHRfoundanupperboundwithalowerobjectivefunctionvalue,

comparedtothevaluereturnedbyCPLEXastheoptimalvalue,whileAlphaECPcouldnotsolvetheproblem.Table4focusesonthesetwoinstances.

#ofVariables Case CHR CPLEX AlphaEcp

400 II 3,358,685 3,359,690 N/A

400 III 49,337,624 49,344,302 N/A

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Asseen,CHRperformsbetterthanCPLEXandAlphaEcpintermsofreturningafeasible

solutionwiththelowestobjectivefunction.However,oneshouldnotethatthisfeasiblesolutionisstillaupperbound,butcannotbeprovenoptimal.

9. CONCLUSIONSANDFUTUREDIRECTIONOFRESEARCH

TheConvexHullRelaxation(CHR)providestightlowerandupperboundsby(1)

transforminganonlinearintegeroptimizationprobleminoneovertheconvexhullofallfeasiblesolutions,and(2)replacingthisproblembyasequenceoflinearprogramsandsimplenonlinearprograms.Thepotentialstrengthoftheproposedalgorithmisthatthedifficultyoftheproblems

solvedateachiterationstaysrelativelyunchangedfromiterationtoiteration.Itwillbemostsuitableforthosenonlinearintegerproblemtypesthatwouldbemucheasiertosolvewithalinearobjectivefunction.OneshouldexpectthatCHRwillhavearobustperformanceforlarge‐scaleproblemsifone

hasaccesstosolversabletohandlelargelinearprogramsandsimplenonlinearprogramsefficiently.Furthertestingisneededforlargerproblemsizesandotherproblemtypes.Themostimportantrestrictionofthemethodseemstobetheconvexityrequirementonthe

objectivefunction,andlinearconstraints.Butthereisnofurtherstructuralrequirementontheobjectivefunction,incontrasttoavailableMINLPsolverssuchasCPLEX,whichrequireaquadraticobjectivefunction.

10. APPENDIX

10.1 AppendixA:ComputationalResultsinDetail

Notethatforthedatasetsusedwehave

Therefore,capacityrequirementscorrespondingtoeachcasehasbeenchosenaccordingly.

A. #ofvariables=100

Table1

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CaseI

(b=5000)

CaseII

(b=125,000)

CaseIII

(b=500,000)

CaseIV

(b=1250,000)

CaseV

(b=2500,000)

Cont.Bound 2649.7 263,678 3,479,767 19,156,699 74,612,738

ConvexHullLowerBound

90377.4 299,457 3,899,468 23,857,310 114,289,200

ConvexHull

UpperBound99766.8 315,028 3,927,058 23,885,029 114,289,580

%Improvement 3310.9 13.57 12.06 24.5 53.2

Optimalvalue(AlphaECP)

99766.8 315,028 3,927,058 23,885,029 114,289,580

Optimalvalue(CPLEX)

99766.8 315,028 3,937,286** 23,885,029 114,289,580

%lowerboundgap

9.41 4.94 0.70 0.12 3.3249e‐004

%upperboundgap

0.0 0.0 0.0 0.0 0.0

**CPLEXWarning:ThesearchwasstoppedbecausetheobjectivefunctionoftheNLPsubproblemstartedtodeteriorate.

Table2‐Runtimesinseconds(b=5,000)

NLP MIP ECP TOTAL

CPLEX 0.88 0.13 ‐ 1.01

AlphaECP ‐ 9.00 1.00 10.00

CHR 0.78 0.29 ‐ 1.07


NLP MIP ECP TOTAL

CPLEX 0.97 0.13 ‐ 1.10

AlphaECP ‐ 8.00 1.00 9.00

CHR 0.43 0.19 ‐ 0.62


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NIP MIP ECP TOTAL

CPLEX 1.02 0.24 ‐ 1.26

AlphaECP ‐ 246.98 4.02 251.00

CHR 1.66 0.780 ‐ 2.440

Table5‐Runtimesinseconds(b=1,250,000)

NIP MIP ECP TOTAL

CPLEX 0.65 0.11 ‐ 0.76

AlphaECP ‐ 28.00 4.00 32.00

CHR 2.10 0.74 ‐ 2.84


NIP MIP ECP TOTAL

CPLEX 0.66 0.17 ‐ 0.83

AlphaECP ‐ 2.00 0.00 2.00

CHR 0.44 0.13 ‐ 0.57

Table8‐AnalysisofSimplicialalgorithm(#ofvariables=100)

bNo.of

SimplicialIterations

Max.noof

extremepointsata

given

iteration

5000 12 12

125000 7 7

500000 15 15

1250000 11 11

2500000 3 3

16

B.#ofvariables=200Table9

**CPLEXWarning:ThesearchwasstoppedbecausetheobjectivefunctionoftheNLPsubproblemstartedtodeteriorate.


NLP MIP ECP TOTAL

CPLEX* 2.31 0.09 ‐ 2.40

AlphaECP ‐ 46.00 7.00 53.00

CHR 2.56 0.39 ‐ 2.95

CaseI

(b=10000)CaseII

(b=250000)CaseIII

(b=1000000)CaseIV

(b=2500000)CaseV

(b=5000000)

Cont.Bound 6181.1 915,114 12,782,780 83,464,883 394253860

ConvexHullLowerbound

92261.1 1,091,412 12,814,670 85,137,220 394,794,400

ConvexHull

UpperBound105,980.6 1,123,458.7 12,820,782 85,158,703 394,794,740

%Improvement 1392.6 19.37 0.25 2.00 0.14

Optimalvalue(AlphaECP)

105,980.6 1,122,090 12,820,782 85,158,703 394,794,740

Optimalvalue(CPLEX)

105,980.6 1,137,485** 12,820,782 85,158,703 394,794,740

%lowerboundgap

14.87 2.64 0.05 0.03 8.6121e‐005

%upperboundgap

0.0 0.12 0.0 0.0 0.0

17


NLP MIP ECP TOTAL

CPLEX 3.73 0.32 ‐ 4.05

AlphaECP 1600 11 1611

CHR 3.42 0.860 ‐ 4.28


NIP MIP ECP TOTAL

CPLEX 2.79 0.19 ‐ 2.98

AlphaECP 1.00 1.00 2.00

CHR 0.89 0.51 ‐ 1.40


NIP MIP ECP TOTAL

CPLEX 2.25 0.34 ‐ 2.59

AlphaECP ‐ 24.00 1.00 25.00

CHR 5.88 0.93 ‐ 6.81


NIP MIP ECP TOTAL

CPLEX 2.19 0.21 ‐ 2.40

AlphaECP ‐ 3.00 0.0 3.00

CHR 1.57 0.21 ‐ 1.78


bNo.of


Max.noof

extremepointsata

given

iteration

18

10000 15 15

250000 16 16

1000000 5 5

2500000 11 11

5000000 3 3

C.#ofvariables=400

Table16

CaseI

(b=20000)CaseII

(b=500000)CaseIII

(b=2000000)CaseIV

(b=5000000)CaseV

(b=10000000)

Cont.Bound 14383 3,275,799 49,203,510 328,628,230 1,661,081,100

ConvexHull

LowerBound91684 3,316,422 49,298,830 329,626,900 1,662,065,000

ConvexHullUpperBound

106,260 3,358,685 49,337,624 329,626,924 1,662,065,750

%Improvement 537.4% 1.24% 0.19% 0.3% 0.06%

Optimalvalue

(AlphaECP)106,260 N/A* N/A* 329,626,924 1,662,065,750

Optimalvalue(CPLEX)

106,260 3,359,690** 49,344,302** 329,626,924 1,662,065,750

19

%lowerbound

gap13.7 1.29 0.09 6.0675e‐006 4.5125e‐005

%upperboundgap

0.0 0.0 0.0 0.0 0.0

*Failtoterminateinlessthan30minutes.**CPLEXWarning:ThesearchwasstoppedbecausetheobjectivefunctionoftheNLPsubproblemstartedtodeteriorate.Table17‐Runtimesinseconds(b=20,000)

NLP MIP ECP TOTAL

CPLEX* 15.00 0.16 ‐ 15.16

AlphaECP ‐ 332.01 43.99 376

CHR 5.18 0.47 ‐ 5.65


NLP MIP ECP TOTAL

CPLEX 18.03 1.51 ‐ 19.54

AlphaECP FAILTOTERMINATE

CHR 17.59 10.20 ‐ 27.79


NIP MIP ECP TOTAL

CPLEX 15.16 0.61 ‐ 15.77

AlphaECP FAILTOTERMINATE

CHR 14.54 126.34 ‐ 140.88


NIP MIP ECP TOTAL

CPLEX 9.95 0.30 ‐ 10.25

AlphaECP ‐ 1.00 3.00 4.00

20

CHR 0.81 0.19 ‐ 1.00


NIP MIP ECP TOTAL

CPLEX 10.55 0.49 ‐ 11.04

AlphaECP ‐ 4.00 3.0 7.00

CHR 7.59 0.53 ‐ 8.12


bNo.of


Max.noofextremepointsata

giveniteration

20000 16 16

500000 30 30

2000000 19 10

5000000 2 2

10000000 4 4

10.2AppendixB:ProofofFactI

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FACTI:

Assumewehavethefollowingmatricesathand:

• A diagonalmatrixDwithpositiveentries,hencepositivedefinite

• A matrixAwithrankm,where

Then ispositivedefinite.

PROOF:BecauseDispositivedefinite,

Hence,thematrixCispositivesemi‐definite.Toshowthatitisalsopositivedefinite,itsufficesto

showthat,.ThisimmediatelyfollowsfromtheassumptionthatAisamatrixofrankm.

10.3AppendixC:GAMSCODEOFTHECHRALGORITHM

$titleGamsCodeforQKPproblemof200variables$eolcom!

optionlimrow=0,limcol=0,solprint=off,sysout=off;$offsymlistoffsymxrefofflistingoptionNLP=minos;

optionoptcr=0;setsijobs/i1*i200/kthisisforsetofextpointsgeneratedforrestsimpdecomp/k1*k1000/

itersetforsimplicialiterations/it1*it5000/ind(k)'indicatoryeswhentheextremepointremains'alias(i,ir);

scalarsroflagflagS

riterationex

effnewfold

minimaxixnormold

normnewfoptimal

counter

22

cccmaster

cccsubccc;variablesy(i)

beta(k)z1z2;

positivevariablesbeta(k);binaryvariablesy(i);

parametersxA(i,iter)u(i)x(i)/i1*i1501/

xold(i)x0(i)t(k)

w(i,k)ayk(i)opt;

equationsobjgradobj

beqncapacityMC;

*****ImportantParameterstoset*****r=200;!'Numberofallowableextremepointsforrestrictedsimplicialdecomposition.100seemsto

befairlysafe,andmakesitactlikeunrestrictedSD.'!'Theseareneededforstoppingcondition.Decreasingthemmaynaturallycausemoreiterations,butimproveaccuracy.'

ef=0.000000000000000000001;ex=0.0000000000000000000001;*===============================================================================

*DATAINPUT=*===============================================================================scalarb/5000000/;

setsj/j1*j500/alias(j,jr);

parametersm(i,j)d(jr,j)q(i,j)

c(i,ir)install(i)a(i);

23

d(jr,j)=0;

m(i,jr)=uniform(0,10);d(jr,jr)=uniform(0,10);q(i,j)=sum(jr,m(i,jr)*d(jr,j));

c(i,ir)=sum(jr,q(i,jr)*m(ir,jr));install(i)=uniform(30000,100000);a(i)=uniform(15000,75000);

;*===============================================================================*EQUATIONS=

*===============================================================================obj..sum(i,install(i)*sum(k$ind(k),beta(k)*w(i,k)))+sum((i,ir),c(i,ir)*sum(k$ind(k),beta(k)*w(i,k))*sum(k$ind(k),beta(k)*w(ir,k)))=e=z2;

objgrad..sum(i,[install(i)+sum(ir,[c(i,ir)+c(ir,i)]*x(ir))]*(y(i)‐x(i)))=e=z1;capacity..sum(i,a(i)*y(i))=g=b;beqn..sum(k$ind(k),beta(k))=e=1;

*===============================================================================*MODELS=*===============================================================================

modelsub/objgrad,capacity/;modelmaster/obj,beqn/;*===============================================================================

*RestoftheInitialization=*===============================================================================*Theseinitializationsarenotmeanttochangeundernormalconditions.

cccmaster=0;cccsub=0;flag=1;

w(i,k)=0;t(k)=0;ind(k)=no;

ind('k1')=yes;fnew=100;fold=0;

maxix=0;flagS=1;iteration=1;

normnew=100;counter=0;

w(i,'k1')=x(i);*===============================================================================*ALGORITHM=

*===============================================================================filecount/count.dat/;fileupperbounds/upperbound.dat/;

24

while(((flagS=1)or((normnew>ex*maxix)and(abs(fnew‐fold)>ef*abs(fold)))),xA(i,iter)$(iteration=ord(iter))=x(i);flagS=0;

solvesubusingmipminimizingz1;cccsub=cccsub+sub.resusd;if((counter<r),

loop(k$((flag=1)and(ord(k)>=2)),if((t(k)=0),w(i,k)=y.l(i);

opt=sum(i,install(i)*y.l(i))+sum((i,ir),c(i,ir)*y.l(i)*y.l(ir));putupperbounds;putopt///;

ind(k)=yes;counter=counter+1;flag=0;

);!endift);!endloopk);!endifcounter

flag=1;if((counter=r),mini=smin(k$(ord(k)>=2andord(k)<r+1),t(k));

loop(k$(ord(k)>=2andord(k)<=r+1),if((mini=t(k)),w(i,k)=y.l(i);

);!endifmini);!endloopkw(i,'k1')=x(i);

);!endwhileflagS=1putcount;putcounter///;

solvemasterusingnlpminimizingz2;cccmaster=cccmaster+master.resusd;t(k)=beta.l(k);

loop(k$(ord(k)>=2andind(k)),if((t(k)=0),

ind(k)=no;);!endifk

);!endloopkxold(i)=x(i);

x(i)=sum(k$ind(k),t(k)*w(i,k));fold=sum(i,install(i)*xold(i))+sum((i,ir),c(i,ir)*xold(i)*xold(ir));fnew=sum(i,install(i)*x(i))+sum((i,ir),c(i,ir)*x(i)*x(ir));

25

normold=sqrt(sum(i,sqr(xold(i))));

normnew=sqrt(sum(i,sqr(x(i))));maxix=max(normnew,normold);iteration=iteration+1;

);!endwhilecounterA=1foptimal=sum(i,install(i)*xold(i))+sum((i,ir),c(i,ir)*xold(i)*x(ir));ccc=cccmaster+cccsub;

displayfoptimal,cccmaster,cccsub,ccc;

11. REFERENCES

Ahn,S.,L.ContesseandM.Guignard,“AnAugmentedLagrangeanRelaxationforNonlinearInteger Programming Solved by theMethod ofMultipliers, Part II: Application toNonlinear FacilityLocation,”WorkingPaper,latestrevision2007.

Bazaraa,M.S.andC.MShetty.,”NonlinearProgrammingTheoryandAlgorithms”,JohnWiley&Sons,Inc.,1979.

BertsekasD.,“NonlinearProgramming”,AthenaScientific,2dprinting,2003.

ContesseL. andM.Guignard, “AnAugmentedLagrangeanRelaxation forNonlinear IntegerProgramming Solved by theMethod ofMultipliers, Part I: Theory andAlgorithm,”Working Paper,OPIMDepartment,Univ.ofPennsylvania,latestrevision2007.

Guignard,M.,’PrimalRelaxationinIntegerProgramming,’VIICLAIOMeeting,Santiago,Chile,1994, also Operations and Information Management Working Paper 94‐02‐01, University ofPennsylvania,1994.

Guignard, M., “A New, Solvable, Primal Relaxation For Nonlinear Integer ProgrammingProblems with Linear Constraints,” Operations and Information Management Working Paper,UniversityofPennsylvania,2007.

Hearn, D.W., Lawphongpanich S. and Ventura J.A, ”Restricted Simplicial Decomposition:ComputationandExtensions”,MathematicalProgrammingStudy31,99‐118,1987.

the convex hull relaxation for nonlinear integer … integer minimization ... while solving a linear...

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