1
TheConvexHullRelaxationforNonlinearIntegerPrograms
WithLinearConstraints
By
AykutAhlatciogluand
MoniqueGuignard1
OPIMDepartmentTheWhartonSchool
UniversityofPennsylvania
Draft20September2007
Donotquotewithouttheauthors’permission
1ResearchpartiallysupportedunderNSFGrantDMI‐0400155.
2
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.
4
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.
5
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.
6
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’
7
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
8
AsanexampleproblemweimplementedCHRtofindalowerboundontheoptimalvalueof
quadraticanti‐knapsackproblems.TheQuadraticAnti‐KnapsackProblem(QKP)isasfollows:
whereisthelinearcostincurredbyselectingitem .
isthequadraticcostincurredbyselectingitems and concurrently.
isthespacefilledifitem isselected.
bisthetotalminimumspaceneededtobefilledintheknapsack.
Followingthediscussionabove,theConvexHullRelaxation(CHR),itssubproblem(IPS)anditsmasterproblem(MP)willbeasfollows:
9
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.
10
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
CaseI CaseII CaseIII CaseIV CaseV
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
CaseI CaseII CaseIII CaseIV CaseV
No.of
Variable
100 9.41 4.94 0.70 0.12
11
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
CaseI CaseII CaseIII CaseIV CaseV
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
12
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
13
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
14
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
Table3‐Runtimesinseconds(b=125,000)
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
Table4‐Runtimesinseconds(b=500,000)
15
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
Table6‐Runtimesinseconds(b=2,500,000)
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.
Table10‐Runtimesinseconds(b=10,000)
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
Table11‐Runtimesinseconds(b=250,000)
NLP MIP ECP TOTAL
CPLEX 3.73 0.32 ‐ 4.05
AlphaECP 1600 11 1611
CHR 3.42 0.860 ‐ 4.28
Table12‐Runtimesinseconds(b=1,000,000)
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
Table13‐Runtimesinseconds(b=2,500,000)
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
Table14‐Runtimesinseconds(b=5,000,000)
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
Table15‐AnalysisofSimplicialalgorithm(#ofvariables=200)
bNo.of
SimplicialIterations
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
Table18‐Runtimesinseconds(b=500,000)
NLP MIP ECP TOTAL
CPLEX 18.03 1.51 ‐ 19.54
AlphaECP FAILTOTERMINATE
CHR 17.59 10.20 ‐ 27.79
Table19‐Runtimesinseconds(b=2,000,000)
NIP MIP ECP TOTAL
CPLEX 15.16 0.61 ‐ 15.77
AlphaECP FAILTOTERMINATE
CHR 14.54 126.34 ‐ 140.88
Table20‐Runtimesinseconds(b=5,000,000)
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
Table21‐Runtimesinseconds(b=10,000,000)
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
Table22‐AnalysisofSimplicialalgorithm(#ofvariables=400)
bNo.of
SimplicialIterations
Max.noofextremepointsata
giveniteration
20000 16 16
500000 30 30
2000000 19 10
5000000 2 2
10000000 4 4
10.2AppendixB:ProofofFactI
21
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$ind(k),beta(k))=e=1;
*===============================================================================*MODELS=*===============================================================================
modelsub/objgrad,capacity/;modelmaster/obj,beqn/;*===============================================================================
*RestoftheInitialization=*===============================================================================*Theseinitializationsarenotmeanttochangeundernormalconditions.
cccmaster=0;cccsub=0;flag=1;
w(i,k)=0;t(k)=0;ind(k)=no;
ind('k1')=yes;fnew=100;fold=0;
maxix=0;flagS=1;iteration=1;
normnew=100;counter=0;
w(i,'k1')=x(i);*===============================================================================*ALGORITHM=
*===============================================================================filecount/count.dat/;fileupperbounds/upperbound.dat/;
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while(((flagS=1)or((normnew>ex*maxix)and(abs(fnew‐fold)>ef*abs(fold)))),xA(i,iter)$(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.
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