Mixed integer polynomial programming

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ContentslistsavailableatScienceDirect

Computers

and

Chemical

Engineering

jo u r n al h om ep age :w w w . e l s e v i e r . c o m / l o c a t e / c o m p c h e m e n g

Mixed

integer

polynomial

programming

Vivek

Dua

∗

DepartmentofChemicalEngineering,CentreforProcessSystemsEngineering,UniversityCollegeLondon,TorringtonPlace,LondonWC1E7JE,United Kingdom

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t

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c

l

e

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Articlehistory:

Received16January2014

Receivedinrevisedform15July2014 Accepted20July2014

Availableonline29July2014 Keywords:

Mixedintegerprogramming Polynomialprogramming Nonlinearinversion

Multi-parametricprogramming

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Themixedintegerpolynomialprogrammingproblemisreformulatedasamulti-parametric program-mingproblembyrelaxingintegervariablesascontinuousvariablesandthentreatingthemasparameters. Theoptimalityconditionsfortheresultingparametricprogrammingproblemaregivenbyasetof simul-taneousparametricpolynomialequationswhicharesolvedanalyticallytogivetheparametricoptimal solutionasafunctionoftherelaxedintegervariables.Evaluationoftheparametricoptimalsolutionfor integervariablesﬁxedattheirintegervaluesfollowedbyscreeningoftheevaluatedsolutionsgivesthe optimalsolutions.

1. Introduction

Mathematical modelling and model-based optimization of chemical process systems have great potential for providing answersonhowtooptimallydesignandoperatethesesystems. Onekeydifﬁcultyinnotbeingabletofullyexploitthispotentialis thepresenceofnonlineartermsinthemathematicalmodels.This issueisfurtherexacerbatedwhenthemodelalsoincludesinteger variables,for incorporatingstructural choices.Ageneric formu-lationofsuchmathematicalprogrammingproblemsisgivenby mixedintegernonlinearprogrammes(MINLP)(Grossmann,2002):

ProblemP1: z1=min x,yf(x,y) subjectto:h(x,y)=0 g(x,y)≤0 x∈nx y∈{0,1}ny

wherexisavectorofcontinuousvariables,yisavectorofbinary variables,hisannhdimensionalvectorofequalityconstraints,g isanngdimensionalvectorofinequalityconstraintsandfisthe scalarobjectivefunction.Synthesisofchemicalprocessﬂowsheets anddesignofmaterialsaretwotypicalproblemsdemonstratingthe

∗ Tel.:+4402076790002;fax:+4402073832348. E-mailaddress:[email protected]

applicationofmathematicalprogrammessimultaneously involv-ingnonlinearities and integervariables (Duaand Pistikopoulos, 1998).Forprocesssynthesisproblems,xrepresentscontinuously varyingquantitiessuchastemperature,pressure,ﬂowratesetc.,y

isusedtomodelstructuraldecisionssuchasselectionof appro-priateprocessingunitsandinter-connectionsbetweentheunits etc.,frepresentsanobjectivefunctionsuchascostor environmen-talimpacttobeoptimized,hrepresentsconservationequations i.e. mass and energy balances and g represents constraints on quantitiessuchaslowestacceptablepurityandhighestallowable safeoperating temperaturesand pressures.For material design problemsxrepresentsmaterialproperties,ymodelsselectionof constituentmoleculargroups,frepresentsdeviationfromdesired propertyvalues,hrepresentspropertypredictioncorrelationsand

grepresentslowerandupperboundsonvaluesof thematerial properties.Notethatthisapproachformaterialdesignproblemsis baseduponmatchingpropertytargetsbutotherformulationsfor suchproblemsalsoexistintheliterature.

SolvingP1isNP-hardandhascreatedhugeinterestfor devel-opingcomputationallyefﬁcientalgorithmsforobtainingsolution ofP1.NewtheoreticaldevelopmentsforsolvingP1havepushed theboundariesof applicationof P1tomany areasin engineer-ingandscience.Severalsoftwaretoolsareavailabletosolvethese problems,DICOPT(ViswanathanandGrossmann,1990),MINOPT (MINOPT,1998),BARON(Sahinidis,1996),GloMIQO(Misenerand Floudas,2013),Alpha-ECP(WesterlundandLundqvist,2005)–to name a few. Thereader is referred torecent survey papers by Belottietal.(2013)andD’AmbrosioandLodi(2011)presenting anoverviewofadvancesforsolvingP1andthebooks(Biegleretal., http://dx.doi.org/10.1016/j.compchemeng.2014.07.020

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1997;Floudas,1995;NemhauserandWolsey,1988)foran intro-ductiontothetopic.

GueddarandDua(2012)reformulatedP1asamulti-parametric nonlinearprogramme(mp-NLP)byﬁrstrelaxingyascontinuous variablesandthentreatingyasparameters.Anapproximate solu-tionofthemp-NLPisobtainedandthatsolutionisthenusedto estimatethesolutionattheterminalnodesoftheBranchandBound (B&B)treeandguidethesearchinthetree;integervariablesarethe branchingvariablesinthetree.Fordetailsofmulti-parametric pro-grammingthereaderisreferredtoDuaandPistikopoulos(1999), Pistikopoulosetal.(2007a,b),Pistikopoulos(2009)and Wittmann-HohlbeinandPistikopoulos(2014).

Inthis work,thecase when f,h andgare polynomial func-tionsisconsidered,problemP1thereforebecomesaMixedInteger PolynomialOptimization(MIPOPT)problem.Patiletal.(2012) pre-sentedaBernsteinpolynomialapproachforsolvingsuchproblems. Telesetal.(2013)proposedadiscretizationapproachusing bilin-eartermsasthebuildingblock.Inthiswork,theintegervariables,

y,arerelaxedascontinuousvariablesandthentreatedas parame-tersresultinginamulti-parametricpolynomialprogramme(mp3). Anexactsolutionoftheresultingmp3canbeobtainedbyexact multi-parametricnonlinearinversionoftheoptimalityconditions, seeforexampleFotiouetal.(2007).Theproposedapproachhence doesnotrequireapproximatesolutionofthemp-NLPfollowedby atreesearchasinGueddarandDua(2012).

In the next section polynomial programming is introduced and an example for exact solution of polynomial programmes ispresented.AnalgorithmforsolvingMIPOPTbasedupon mp3 reformulationisproposedinSection3andinSection4 illustra-tiveexamplesarepresented.Adiscussionofresultsandconcluding remarksareprovidedinSection5.

2. Polynomialprogramming

Considerthefollowingnonlinearprogramming(NLP)problem: ProblemP2: z2=min x f(x) subjectto:h(x)=0 g(x)≤0 x∈nx

DescentorsimilaralgorithmsforcomputingsolutionofP2are baseduponaniterativestrategy wherethesolutionobtainedat aniterationveriﬁesFritz–John(FJ)orKarush–Kuhn–Tucker(KKT) conditions(Bazaraaetal.,1993),inthisworktheKKTconditions areconsidered,asfollows.

KKTconditions: (a)Equalityconstraints:

∇

xL(x,␮,␭)=0 h(x)=0 jgj(x)=0,j=1,...,ng (b)Inequalityconstraints: g(x)≤0 j≥0,j=1,...,ng where L(x,␮,␭)=f(x)₊ n_h

i=1 ihi(x)₊ ng

j=1 jgj(x) istheLagrangianfunction.

Fig.1.Example1,plotoftheobjectivefunction,f,asafunctionofx1andx2.

TheequalityconstraintsintheKKTconditionsarenx+nh+ng dimensionalandthevectorofvariables,[x,␮,␭],isalsonx+nh+ng dimensional.Forgenericnonlinearfunctionssolutionofequality constraintsis usuallyobtainedby employinga numerical tech-nique, such as Newton’s method. The solution of the equality constraintsobtainedisverifiedbycheckingwhetheritsatisfiesthe inequalityconstraintsintheKKTconditions.Consideringaspecial casewhenf,gandhintheNLParepolynomial,asetofequations polynomialin[x,␮,␭]isobtained.Thesepolynomialequationscan besolvedanalytically,atleastintheory,toobtainaclosedform solutionwhichincludesallthesolutions(Hägglöfetal.,1995).This canbeachievedbyusingthetheoryofGröbnerBaseswherethe Buchbergeralgorithmcanbeusedtotransformthesetof polyno-mialequationsintoatriangularsystemofequations(Buchberger andWinkler,1998).Thetriangularsystemisthenonlinear poly-nomialequivalentofthetriangularsystemobtainedbyGaussian eliminationfora linearsystemofequations. Thecomputational complexityofthismethodgrowsexponentiallywiththenumber ofvariables,butitisanactiveareaofresearchwithvarious devel-opmentsincludingparallelcomputingtoimprovecomputational speed. Thereare softwaresfor symbolic manipulationssuchas Mathematica(WolframResearch,2013)thatcananalyticallysolve systemsofpolynomialequations,whichinourcaseisgivenbythe equalityconstraintsintheKKTconditions.Thesetofsolutionsthus obtainedcanthenbecheckedtoseeiftheysatisfytheinequality constraintsintheKKTconditions.Furtherscreeningtestsarealso carriedout,i.e.,whethercomplementaryslackness(CS)and con-straintsqualification(CQ)conditionsaremetischecked.In this paperlinearindependenceconstraintqualification(LICQ)wasused forcheckingtheCQcondition.Considerthefollowingillustrative examplefordemonstratingthebasicidea.

Example1:Polynomialprogramming min x f(x)=5x 2 1+9x22−8x1x2 subjectto: g1(x)=2−x1x2≤0 g2(x)=−3+x1x2≤0 g3(x)=−5+x1≤0 g4(x)=−5−x1≤0

Theobjectivefunctionisplottedasafunctionofx1 andx2 in Fig.1andthefeasibleregionisgivenbytheshadedareainFig.2. ThetwooptimalsolutionsarealsoshowninFig.2.

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Fig.2.Example1,plotsofg1=0andg2=0onx1−x2axes.Thefeasibleregion

isgivenbytheshadedarea.Theoptimalsolutionsaregivenbythetwoﬁlled circles(䊉).

TheLagrangianfunctionisgivenby: L(x,␭)=5x2

1+9x22−8x1x2+1(2−x1x2)+2(−3+x1x2) +3(−5+x1)+4(−5−x1)

TheKKTconditionsaregivenby: (a)Equalityconstraints:

∂L ∂x1 = 10x1−8x2+1(−x2)+2(x2)+3−4=0 ∂L ∂x2 =18x2−8x1+1(−x1)+2(x1)=0 1(2−x1x2)=0 2(−3+x1x2)=0 3(−5+x1)=0 4(−5−x1)=0 (b)Inequalityconstraints: 2−x1x2≤0 −3+x1x2≤0 −5+x1≤0 −5−x1≤0 1,2,3,4≥0

Table1showsallthefifteensolutionsoftheKKTequality con-straintsasobtainedbyusingtheSolvecommandinMathematica. Solutionnumbers1–3,5–8,11and13–14areignoredbecauseat leastone Lagrangemultiplier isnegative, solutionnumber 4 is ignoredbecausetheinequalityconstraintg1isviolated,solution numbers9and10areignoredbecausetheyhaveimaginaryparts andhencesolutionnumbers12and15arethetwocandidate solu-tions.Notethatconsideringboththepositiveandnegativevalues ofthesquareroots,theLagrangemultipliersofthe8thand11th solutionscouldtakepositivevalues.Thesesolutionshoweverget rejectedlaterbecauseofthehigherobjectivefunctionvaluesthan thebestidentified.Solutions12and15werefurthertestedand satisfiedCSandCQandhencearethetwofinalsolutions,asalso showninFig.2.

3. Analgorithmformixedintegerpolynomialoptimization

(MIPOPT)usingmulti-parametricpolynomialprogramming

(mp3)

RecallproblemP1andnowconsiderthecasethatf,handg

arepolynomialfunctionsofxandthatforsimplicitythetermsin

xandyareseparable,thisresultsinaMixedIntegerPolynomial Optimization(MIPOPT)problem.Relaxingtheintegervariablesas continuousvariablesandtreatingthemasparametersresultsinthe followingmulti-parametricpolynomialprogramme(mp3):

ProblemP3: z3(y)=min x f(x,y) subjectto:h(x,y)=0 g(x,y)≤0 x∈nx y∈ [0,1]ny

TheKKTconditionsforthemp3areasfollows. KKTconditionsforProblemP3:

(a)Equalityconstraints:

∇

xL(x,y,␮,␭)=0 h(x,y)=0 jgj(x,y)=0,j=1,...,ng (b)Inequalityconstraints: g(x,y)≤0 j≥0,j=1,...,ng where L(x,y,␮,␭)=f(x,y)+ n_h

i=1 ihi(x,y)+ ng

j=1 jgj(x,y) istheLagrangianfunctionwhichisparametriciny.

TheequalityconstraintsintheKKTconditionsarenx+nh+ng dimensional,the vector ofvariables [x,␮,␭] is alsonx+nh+ng dimensionalandthevectorofparameters, y,is ny dimensional. Theseequalityconstraintswhichareparametricinyand polyno-mialin[x,␮,␭]aresolvedanalyticallye.g.byusingMathematica. Theparametricsolutionisgivenby[x(y),␮(y),␭(y)].The parame-ters,y,arethenﬁxedatpossibleintegervaluesandsubstitutedinto theparametricsolutiontoevaluate[x(y),␮(y),␭(y)].Thesevalues arethensubstitutedintotheinequalityconstraintsintheKKT con-ditions.Thesolutionswhichsatisfytheseinequalityconstraints,CS andCQarethensubstitutedintotheobjectivefunction,f(x,y),and thesolutionswhichgivethelowestvaluesaretheﬁnalsolutions. ThealgorithmissummarizedinTable2.

Forsomespeciﬁcproblemsvariationsoftheproposedalgorithm canalsobedeveloped.Forexample,afterstep4thevaluesofthe continuousvariables andtheLagrangemultipliersare available. Thereforeinstep5thesolutionswhichprovidenegativeLagrange multipliersforinequalityconstraintscanbeignoredfromfurther analysisand satisfactionofotherKKT inequalityconstraints,CS andCQisnotrequiredtobecarriedoutforthosesolutions.The solutionswhichgiveimaginaryvaluesforthecontinuousvariables orLagrangemultiplierscanalsoberemoved.Fortheremaining candidatesolutionstwopossibleoptionsareasfollows.

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Table1

Example1,solutionofequalityconstraintsintheKKTconditions.Notethatidenotesthecomplexpartofthesolution.

⎧

⎪

⎨

⎪

⎩

(1) 1=0 2=0 3=0 4=−290 9 x1=−5 x2=− 20 9 (2) 1=0 2= 146 25 3=0 4= 6088 125 x1=−5 x2=− 3 5 (3) 1=− 164 25 2=0 3=0 4= 6178 125 x1=−5 x2=− 2 5 (4) 1=0 2=0 3=0 4=0 x1=0 x2=0 (5) 1=0 2=146 25 3=− 6178 125 4=0 x1=5 x2= 2 5 (6) 1=0 2= 146 25 3= 6088 125 4=0 x1=5 x2= 3 5 (7) 1=0 2=0 3=290 9 4=0 x1=5 x2= 20 9 (8) 1=0 2=2(4−3 √ 5) 3=0 4=0 x1=− 3 51/4 x2=−5 1/4 (9) 1=0 2=2(4+3 √ 5) 3=0 4=0 x1=− 3i 51/4 x2=−i5 1/4 (10) 1=0 2=2(4+3 √ 5) 3=0 4=0 x1= 3i 51/4 x2=−i5 1/4 (11) 1=0 2=2(4−3 √ 5) 3=0 4=0 x1= 3 51/4 x2=5 1/4 (12) 1=2(−4+3 √ 5) 2=0 3=0 4=0 x1= √ 6 51/4 x2=−

2 35 1/4 (13) 1=2(−4−3 √ 5) 2=0 3=0 4=0 x1=− i√6 51/4 x2=i

2 35 1/4 (14) 1=2(−4−3 √ 5) 2=0 3=0 4=0 x1= i√6 51/4 x2=−i

2 35 1/4 (15) 1=2(−4+3 √ 5) 2=0 3=0 4=0 x1= √ 6 51/4 x2=

2 35 1/4

⎫

⎪

⎬

⎪

⎭

Table2 MIPOPTalgorithm.

Step1. ReformulateMIPOPTproblemasamulti-parametric polynomialprogramme,byrelaxingintegervariablesas continuousvariablesandtreatingthemasparameters,as giveninProblemP3.

Step2. FormulateﬁrstorderKKTConditionsforProblemP3,given byEqualityConstraintsandInequalityConstraints. Step3. SolvetheEqualityConstraintsintheKKTConditions

parametricallytoobtaincontinuousvariablesand Lagrangemultipliersasfunctionofrelaxedinteger variables.

Step4. Fixtheintegervariablesatallthepossibleintegervalue combinationsandevaluatethecontinuousvariablesand Lagrangemultipliersobtainedinthepreviousstep. Step5. Screenthesolutionsobtainedinthepreviousstepto

obtainthebestsolution(s).Thisinvolvescheckingwhether theInequalityConstraintsintheKKTConditions, ComplementarySlackness(CS)andConstraint Qualiﬁcation(CQ)andaresatisﬁedandidentifyingthe solutionswhichgivethelowestvalueoftheobjective function.

• Theﬁrstsolutionwhich satisﬁesalltheconditions actsasan

upperboundand the remainingsolutions which have higher

objectivefunctionvalue,even withoutcheckingwhetherthey

satisfy any conditions or constraints, are ignored. The upper

boundisthenupdatedasbetteroptimalsolutionsareidentiﬁed.

• Thesolutionswhichviolateevenoneinequalityconstraintare

removed.TheremainingsolutionsarethentestedforCS,CQand

lowestobjectivefunctionvalue.

4. Numericalexamples

Thissection presents four examples in detail. The ﬁrst two

examplesaretakenfromFloudasetal.(1999)andhavealsobeen

solvedbyPatiletal.(2012).Thelasttwoexamplesaretakenfrom

FloudasandPardalos(1990)andalsosolvedbyLasserre(2001),and havebeenmodiﬁedforformulatingMIPOPTproblems.

4.1. Example2:MIPOPT

Considerthefollowingproblem: minf x,y =y1+y2+y3+5x 2 1 subjectto: g1=3x1−y1−y2≤0 g2=−x1−0.1y2+0.25y3≤0 g3=0.2−x1≤0

wherex1 isthecontinuousvariableandy1,y2,y3 arethe0–1 binaryvariables. Byrelaxingthebinaryvariables ascontinuous variablesandthentreatingthemasparametersanmp3isobtained. Forsimplicitythelowerandupperboundsonthebinaryvariables areignoredatthisstage.TheLagrangianfunctionisgivenby: L=y1+y2+y3+5x12+1(3x1−y1−y2)

+2(−x1+0.1y2+0.25y3)+3(0.2−x1) TheKKTconditionsaregivenby:

∂L

∂x1 =10x1+31−2−3=0 1(3x1−y1−y2)=0

2(−x1+0.1y2+0.25y3)=0 3(0.2−x1)=0

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(b)Inequalityconstraints:

3x1−y1−y2≤0 −x1+0.1y2+0.25y3≤0 0.2−x1≤0

1,2,3≥0

Solving the equality constraints in the KKT conditions by relaxingtheintegervariables ascontinuousvariables and treat-ingthem asparameters, thefollowingparametricsolutions are obtained:

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⎨

⎪

⎩

(1) {1=0,2=0,3=2,x1=0.2} (2) _{1₌0,2=0,3=0,x1=0_}

(3) {1=0,2=y2+2.5y3,3=0,x1=0.1y2+0.25y3} (4) {1=−1.11(y1+y2),2=0,3=0,x1=0.33(y1+y2)}

⎫

⎪

⎬

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Remark1. Notethatthesesolutionsareparametricin[y1,y2,y3]

andnoboundswereimposedon[y1,y2,y3]whileobtainingthese solutions.Thesolutionthusobtainedandthemethodology pro-posedinthisworkisthereforealsovalidforthegenericcasewhere

ycanbeanyintegervariableandnotjustlimitedtothe0–1binary values.SeeSections4.3and4.4forthegenericmixed-integercase examples.

Nowthesesolutionsarescreenedbyﬁxingthebinaryvariables atthe0–1valuesandsubstitutingthevaluesintothesesolutions and thenchecking theinequalityconstraintsin theKKT condi-tions,CSandCQconditions.Thefollowingtwosolutionspassthe screeningtestandsincethesecondsolutiongivesthelowest objec-tivefunctionvalue,f=2.20,itistheﬁnalsolution.

1 2 3 x1 y1 y2 y3 f 0 2.50 0 0.25 1.00 0 1.00 2.31 0 0 2.00 0.20 1.00 1.00 0 2.20 4.2. Example3:MIPOPT min x,yf = 2y1+2y2+4x1−x 2 1−x22+2x2+2 subjectto: g1 = −x1+3x2−5≤0 g2=2x1−x2−5≤0 g3=−2x1+x2≤0 g4=x1−3x2≤0 g5=−6y1+x1≤0 g6=−5y2+x2≤0

wherex1andx2arethecontinuousvariableandy1,y2arethe 0–1binaryvariables.TheLagrangianfunction,byrelaxingthe inte-gervariablesascontinuousvariables,treatingthemasparameters andbyignoringthelowerandupperboundsonthebinaryvariables, isgivenby:

L = 2y1+2y2+4x1−x12−x22+2x2+2+1(−x1+3x2−5) +2(2x1−x2−5)+3(−2x1+x2)+4(x1−3x2) +5(−6y1+x1)+6(−5y2+x2)

TheKKTconditionsaregivenby: (a)Equalityconstraints:

∂L ∂x1 =−2x1−1+22−23+4+5+4=0 ∂L ∂x2 =−2x2+ 31−2+3−34+6+2=0 1(−x1+3x2−5)=0 2(2x1−x2−5)=0 3(−2x1+x2)=0 4(x1−3x2)=0 5(−6y1+x1)=0 6(−5y2+x2)=0 (b)Inequalityconstraints: −x1+3x2−5≤0 2x1−x2−5≤0 −2x1+x2≤0 x1−3x2≤0 −6y1+x1≤0 −5y2+x2≤0 1,2,3,4,5,6≥0

BysolvingtheequalityconstraintsintheKKTconditionsthe solutionsobtainedaregiveninTable3.Thesolutionsobtainedafter thescreeningtestforvaluesofyfixedatintegervaluesaregiven inTable4.Thefirstsolutiongivesthelowestpossiblevalueoffand henceisthefinalsolution.

4.3. Example4:MIPOPT min x,yf = −25(x1−2) 2 −(x2−2)2−(x3−1)2−(y1−4)2 −(y2−1)2−(y3−4)2 subjectto: g1 = −(x3−3)2−y1+4≤0 g2=−(y2−3)2−y3+4≤0 g3=x1−3x2−2≤0 g4=−x1+x2−2≤0 g5=x1+x2−6≤0 g6=−x1−x2+2≤0 x1,x2≥0;1≤x3≤5;0≤y1≤6;1≤y2≤5;0≤y3≤10

where x1,x2, x3 arethecontinuousvariables and y1, y2,y3 are theintegervariables.Forsimplicity,ignoringthesimplelowerand upperboundsonthevariables,theLagrangianfunctionisgivenby: L= −25(x1−2)2−(x2−2)2−(x3−1)2−(y1−4)2−(y2−1)2

−(y3−4)2+1(−(x3−3)2

−y1+4)+2(−(y2−3)2−y3+4) +3(x1−3x2−2)+4(−x1+x2−2)+5(x1+x2−6) +6(−x1−x2+2)

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Table3

Example3,parametricsolutionofequalityconstraintsinKKTconditionsforintegervariablesrelaxedascontinuousvariablesandtreatedasparameters.

1 2 3 4 5 6 x1 x2 1 0 0 0 0 0 0 2 1 2 0 0 0 0 0 2(−1+5y2) 2 5y2 3 0 0 0 0 4(−1+3y1) 0 6y1 1 4 0 0 0 0 4(−1+3y1) 2(−1+5y2) 6y1 5y2 5 0 0 0 1/5 0 0 21/10 7/10 6 0 0 0 (−2/3)(−1+2y1) (2/3)(−7+20y1) 0 6y1 2y1 7 0 0 0 2(−2+15y2) 0 2(−7+50y2) 15y2 5y2

8 0 0 6/5 0 0 0 4/5 8/5

9 0 0 14/5 8/5 0 0 0 0

10 0 0 2(−1+12y1) 0 4(−2+15y1) 0 6y1 12y1 11 0 0 (1/2)(4−5y2) 0 0 (1/2)(−8+25y2) 5y2/2 5y2 12 0 4/5 0 0 0 0 14/5 3/5 13 0 6/5 0 −2/5 0 0 3 1 14 0 −12(−1+2y1) 0 0 4(−7+15y1) 0 6y1 −5+12y1 15 0 (1/2)(1+5yy2) 0 0 0 (1/2)(−3+25y2) (5(1+y2))/2 5y2 16 2/5 0 4/5 0 0 0 1 2 17 4/5 0 0 0 0 0 8/5 11/5 18 12/5 16/5 0 0 0 0 4 3 19 (4/9)(1+3y1) 0 0 0 (8/9)(−4+15y1) 0 6y1 (1/3)(5+6y1) 20 −2(−7+15y2) 0 0 0 0 4(−11+25y2) 5(−1+3y2) 5y2

wheretheintegervariablesarerelaxedascontinuousvariablesand

treatedasparameters.TheKKTconditionsaregivenby:

∂L ∂x1 =−50(x1− 2)+3−4+5−6=0 ∂L ∂x2 =−2(x2− 2)−33+4+5−6=0 ∂L ∂x3 =−2(x3− 1)−21(x3−3)=0 1(−(x3−3)2−y1+4)=0 2(−(y2−3)2−y3+4)=0 3(x1−3x2−2)=0 4(−x1+x2−2)=0 5(x1+x2−6)=0 6(−x1−x2+2)=0 (b)Inequalityconstraints: −(x3−3)2−y1+4≤0 −(y2−3)2−y3+4≤0 x1−3x2−2≤0 −x1+x2−2≤0 x1+x2−6≤0 −x1−x2+2≤0 1,2,3,4,5,6≥0

SolvingtheequalityconstraintsintheKKTconditionsthe

solu-tionsobtainedaregiveninTable5.Evaluatingthesesolutionsfor

yﬁxedatintegervaluesandscreeningthesesolutionstheoptimal solutionisgivenbyx=[5,1,5],y=[0,5,10],f=−310.Notethatwhen evaluatingthesolutionsinTable5bothpositiveandnegativesigns

ofthesquare rootvalues mustbeconsidered.g2 isapure inte-ger/parametricconstraintandcanbetakenoutoftheinitialKKT analysisandbethenusedlaterinthescreeningtest.

4.4. Example5:MIPOPT min x,yf = −12x1−7y1+y 2 1 subjectto: h1 =−2x41−y1+2=0 0≤x1≤2;0≤y1≤3

wherex1isthecontinuousvariableandy1istheintegervariable. Forsimplicity,ignoringthesimplelowerandupperboundsonthe variables,theLagrangianfunctionisgivenby:

L= −12x1−7y1+y21+1(−2x41−y1+2)

wheretheintegervariableisrelaxedasacontinuousvariableand treatedasaparameter.TheKKTconditionsaregivenby:

∂L

∂x1 =−81x 3

1−12=0 h1 =−2x41−y1+2=0

SolvingtheequalityconstraintsintheKKTconditionsthe solu-tionsobtainedaregivenby:

1 x1 1 3 21/4_(2−y 1)3/4 −(2−y1)1/4 21/4 2 3i 21/4_(2−y 1)3/4 −i(2−y1)1/4 21/4 3 − 3i 21/4_(2−y 1)3/4 i(2−y1)1/4 21/4 4 − 3 21/4_(2−y 1)3/4 (2−y1)1/4 21/4

whereiinthe2ndand3rdsolutionisthecomplexpartofthe solu-tion,the2ndand3rdsolutionsarethusignored.Evaluating1stand 4thsolutionsforyﬁxedatintegervaluesandscreeningthese solu-tionstheoptimalsolutionisgivenbyx1=0.84,y1=1,1=−2.52, f=−16.09.

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Table4

Example3,setofcandidatesolutionssatisfyingtheinequalityconstraintsintheKKTconditions.

1 2 3 4 5 6 x1 x2 y1 y2 f 1 0.0 0.0 2.8 1.6 0.0 0.0 0.0 0.0 0.0 0.0 2.0 2 0.0 0.0 2.8 1.6 0.0 0.0 0.0 0.0 0.0 1.0 4.0 3 0.0 0.0 2.8 1.6 0.0 0.0 0.0 0.0 1.0 0.0 4.0 4 0.0 0.0 0.0 0.0 0.0 0.0 2.0 1.0 1.0 1.0 11.0 5 0.0 0.0 0.0 0.2 0.0 0.0 2.1 0.7 1.0 1.0 10.9 6 0.0 0.0 1.2 0.0 0.0 0.0 0.8 1.6 1.0 1.0 9.2 7 0.0 0.0 2.8 1.6 0.0 0.0 0.0 0.0 1.0 1.0 6.0 8 0.4 0.0 0.8 0.0 0.0 0.0 1.0 2.0 1.0 1.0 9.0 9 0.8 0.0 0.0 0.0 0.0 0.0 1.6 2.2 1.0 1.0 9.4 10 2.4 3.2 0.0 0.0 0.0 0.0 4.0 3.0 1.0 1.0 3.0 Table5

Example4,parametricsolutionofequalityconstraintsinKKTconditionsforintegervariablesrelaxedascontinuousvariablesandtreatedasparameters. 1 2 3 4 5 6 x1 x2 x3 1 0 0 0 0 0 0 2 2 1 2 0 0 0 0 0 50/13 25/13 1/13 1 3 0 0 0 0 50/13 0 27/13 51/13 1 4 0 0 0 50/13 0 0 25/13 51/13 1 5 0 0 0 50 0 50 0 2 1 6 0 0 0 2 2 0 2 4 1 7 0 0 150/113 0 0 0 229/113 1/113 1 8 0 0 1 0 0 1 2 0 1 9 0 0 38 0 112 0 5 1 1 10 0 0 154 454 0 0 −4 −2 1 11 (4−2

4−y1−y1)/(−4+y1) 0 0 0 0 0 2 2 3−

4−y1 12 (4−2

4−y1−y1)/(−4+y1) 0 0 0 0 50/13 25/13 1/13 3−

4−y1 13 (4−2

4−y1−y1)/(−4+y1) 0 0 0 50/13 0 27/13 51/13 3−

4−y1 14 (4−2

4−y1−y1)/(−4+y1) 0 0 50/13 0 0 25/13 51/13 3−

4−y1 15 (4−2

4−y1−y1)/(−4+y1) 0 0 50 0 50 0 2 3−

4−y1 16 (4−2

4−y1−y1)/(−4+y1) 0 0 2 2 0 2 4 3−

4−y1 17 (4−2

4−y1−y1)/(−4+y1) 0 150/113 0 0 0 229/113 1/113 3−

4−y1 18 (4−2

4−y1−y1)/(−4+y1) 0 1 0 0 1 2 0 3−

4−y1 19 (4−2

4−y1−y1)/(−4+y1) 0 38 0 112 0 5 1 3−

4−y1 20 (4−2

4−y1−y1)/(−4+y1) 0 154 454 0 0 −4 −2 3−

4−y1 21 (4+2

4−y1−y1)/(−4+y1) 0 0 0 0 0 2 2 3+

4−y1 22 (4+2

4−y1−y1)/(−4+y1) 0 0 0 0 50/13 25/13 1/13 3+

4−y1 23 (4+2

4−y1−y1)/(−4+y1) 0 0 0 50/13 0 27/13 51/13 3+

4−y1 24 (4+2

4−y1−y1)/(−4+y1) 0 0 50/13 0 0 25/13 51/13 3+

4−y1 25 (4+2

4−y1−y1)/(−4+y1) 0 0 50 0 50 0 2 3+

4−y1 26 (4+2

4−y1−y1)/(−4+y1) 0 0 2 2 0 2 4 3+

4−y1 27 (4+2

4−y1−y1)/(−4+y1) 0 150/113 0 0 0 229/113 1/113 3+

4−y1 28 (4+2

4−y1−y1)/(−4+y1) 0 1 0 0 1 2 0 3+

4−y1 29 (4+2

4−y1−y1)/(−4+y1) 0 38 0 112 0 5 1 3+

4−y1 30 (4+2

4−y1−y1)/(−4+y1) 0 154 454 0 0 −4 −2 3+

4−y1

5. Discussionofresultsandconcludingremarks

TheproposedMIPOPTtomp3reformulationfollowedbyexact

solutionofthemp3providesallthepossiblecandidatesolutions

anddoesnotuseaniterativenumericalmethodinthetraditional

sensetoidentifytheglobaloptimalsolutions.Inthiswork

Math-ematica was used for obtainingthe parametric solution of the

parametricpolynomialequations,obtainedfromtheKKT

condi-tionsofmp3,howeverothertoolsarealsoavailableforcarrying

(8)

toimprovethetimerequiredtosolvetheparametricpolynomial

equationswillhelpincreasethecomputationalefﬁciencyofthe

proposedalgorithm.Althoughthefocusherewasonthe

nonlinear-itiesarisingonlyfromthepolynomialterms,theoreticalresearch

workonobtaininganalyticalsolutionofequationsincorporating

othertypesofnonlinearitieswillfurtherexpandtheapplicability

oftheproposedapproachwheremixedintegerprogrammeswere

tobereformulatedasmulti-parametricprogrammes.Additionally,

manynonlinearfunctionscanbeapproximatedaspolynomialsand

inthosecasestheproposedapproachcanbeexploredforobtaining

approximateglobaloptimalsolutions.

Acknowledgments

ThispaperisdedicatedtoProfessorIgnacioGrossmannin

cel-ebration of his 65th birthday and in acknowledgement of his

contributionstotheﬁeldofProcessSystemsEngineering.

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