Proc.,FirstASMOUK/ISSMOCONFERENCEonEngineeringDesignOptimization,July8–9,1999,MCBPress COMPARATIVE PROPERTIES OFCOLLABORATIVE OPTIMIZATION ANDOTHER APPROACHESTO MDO NataliaM.Alexandrov andRobertMichaelLewis (cid:3) y MultidisciplinaryOptimizationBranch,MailStop159 (cid:3) InstituteforComputerApplicationsinScienceandEngineering,MailStop132C y NASALangleyResearchCenter,Hampton,Virginia23681-2199,USA E-mail: [email protected], [email protected] (cid:3) y ABSTRACT Thisdistinctioniscrucial,althoughitisoftenblurredin presentations of new approaches to MDO. An analysis We discuss criteria by which one can classify, analyze, ofanMDOformulationconsiderssuchattributesascon- andevaluateapproachestosolvingmultidisciplinaryde- sistency, well-posedness, equivalence to other formula- signoptimization(MDO)problems. Centraltoourdis- tions,optimalityconditions,and sensitivityofsolutions cussionistheoftenoverlookeddistinctionbetweenques- tovariousperturbations. Ananalysisofanoptimization tionsofformulatingMDOproblemsandsolvingthere- algorithm for solving a given formulation of an MDO sultingcomputationalproblem.Weillustrateourgeneral problem then considers local convergence rates, global remarks bycomparing several approaches toMDO that convergence properties, and iteration costs. This work havebeenproposed. discussesthepropertiesofMDOformulations,including INTRODUCTION theireffectonoptimizationalgorithms. AsizablebodyofapproachestosolvingMDOprob- Therearelikelyasmanydefinitionsofmultidisciplinary lemshasbeenproposedovertheyears. However,thereis designoptimization(MDO)asthereareareasandphases asyetonlylimitedcomputationaloranalyticalsubstanti- of design. For our discussion, we shall take MDO to ationofthepracticalapplicabilityandalgorithmicprop- mean the systematic approach to optimization of com- ertiesoftheproposedmethods. Anumber ofrecent ef- plex, coupled engineering systems, where “multidisci- forts(e.g.,AlexandrovandKodiyalam,1999)havebeen plinary” refers to the different aspects that must be in- aimed at addressing this deficiency. The present work cludedinadesign problem. For instance, thedesignof pursuesthefollowingobjectives: aircraft involves, among other disciplines, aerodynam- ics, structural analysis, propulsion, and control. See Theenunciationofasystematicsetofcriteriafor Sobieszczanski-SobieskiandHaftka(1997),Alexandrov (cid:15) analyticalandpracticalevaluationofMDOmeth- andHussaini(1997)foroverviewsofthefield. ods; Broadly speaking, in engineering design problems oneattemptstoimproveoroptimizeseveralobjectives— TheclassificationofMDOformulationsaccording frequentlycompetingandconflictingmeasuresofsystem (cid:15) totheapproachtomaintainingfeasibilitywithre- performance—subject to satisfying a set of design and specttoanalysisanddesignconstraints; physical constraints. It is the nature of some of these The analysisof several formulationsaccording to constraintsthatdistinguishestheengineeringdesignop- (cid:15) theaforementionedset ofcriteriainordertogive timizationproblemfromtheconventionalnonlinearpro- some understanding of the trade-offs among the gramming problem (NLP). As we discuss, the method variousformulations; oftreatingtheproblemconstraintsprovidesthedefining characteristics for various approaches to solving MDO Asketchofhowfeaturesofsomespecificformu- problems. (cid:15) lations of MDO problems affect optimization al- The problem solutiontechniques comprise two ma- gorithms. jorelements: posingtheproblem as aset of mathemat- ical statements amenable to solution and then defining The paper is organized according to this program. We a procedure for solving the problem once it has been hopetoprovidethereadersomeguidancetounderstand- posed. Weusetheterm“formulation”todenotethefirst ing the algorithmic and performance consequences of element and theterm “algorithm”todenotethesecond. choosingoneformulationoranotherforsolvinganMDO 1 Proc.,FirstASMOUK/ISSMOCONFERENCEonEngineeringDesignOptimization,July8–9,1999,MCBPress problem.Amoredetailedandcomprehensivediscussion ematicallyequivalent,doesimprovementinthereformu- canbefoundinAlexandrovandLewis(1999b). lationatleastcorrespondtoimprovementintheoriginal problem? CHARACTERISTICSOFMDOFORMULATIONS Equally important notions of equivalence are more By MDO we mean that subset of the total design subtle. For instance, there is the question of how op- problem—probably in the conceptual or preliminary timalityconditions, constraint qualifications, and sensi- phase—thatcanbeformulatedasanNLPoftheform: tivitycalculationsin(1)correspondtothoseinanalter- nativeformulation. These other notionsof equivalence minimize havebearingonthepracticalapplicationofoptimization (1) subjectto f(x;u(x)) algorithmsto the solutionof formulationsof the MDO g(x;u(x))(cid:21)0; problem. A formulationmay be mathematically equiv- where isthevectorofdesignvariablesand isde- alent to (1), yet algorithms applied to its solution may finedvxiaablocksystemofequations, u(x) exhibitdrasticallydifferentbehavior than when applied to(1),andmayevenfailinsomecases. Wewilltouchon A1(x;u1(x);.:::;uN(x)) thispointagaininourdiscussionofCollaborativeOpti- . 0 . 1 mization;furtherdetailscanbefoundinAlexandrovand A(x;u(x))= =0; Lewis(1999a). B C @ AN(x;u1(x);:::;uN(x)) A (2) beingthenumberofblocks. InthecontextofMDO, EaseofImplementation Ntheblocksofthesystemusuallyrepresentthestateequa- Typically,atremendousamountoftimeandeffortisre- tionsforthedisciplinaryanalyses and thenecessary in- quiredtointegratetheanalysis softwareneeded for any terdisciplinarycouplings. The state equations normally givenformulationofMDOanditssolution.Inparticular, formaset ofcoupleddifferentialequations. System(2) atpresentthereislittleMDAcapabilityinexistingsoft- isknownas theMultidisciplinaryAnalysis(MDA) sys- ware, and adding this capabilityrequires a lotof work. tem. We have simplifiedthe problem by assuming that Thisleadstothenextconsideration. themultipleobjectives ofthesystem have been synthe- sizedinasingleobjective , becausemostextantMDO MultidisciplinaryAnalysis formulationsmakethisassufmption. Ateachiterationofaconventionaloptimizationpro- MDA is expensive and requires aconsiderableeffort to cedure,thedesignvariablevector ispassedtotheMDA implement.Thetypicalapproachtoavoidingtheexpense system. Thesystemisthensolvedxforthestatevector . of an explicit MDA is to introducea relaxation of this Thisreducesthedimensionoftheoptimizationprobleum system, examples of which we will discuss later. One (1)bymakingitaproblemin only. However,eachdis- doesrequiresthatthefullset ofMDAequationsbesat- ciplinary analysis may involvxean expensive procedure, isfiedasoneapproachesanoptimaldesign. say, solvinga differentialequation. Moreover, tosolve Ontheotherhand,sincetheMDAunderliestheorig- theentireMDAsystemonehastouseaniterativeproce- inal problem (1), any attempt to avoid an MDA as an durethatbringstheindividualanalysesintoamultidisci- explicit calculation enforced at each step of the opti- plinaryequilibrium. mization must turn some or all of the MDA equations It is the expense of implementing and executing a intoconsistencyconstraintsintheresultingoptimization straightforward, conventional optimization approach to problem. (Provided,ofcourse, thattheresultingformu- (1)thathas mainlymotivatedresearchers toproposeal- lationisequivalenttotheoriginalproblem(1).) Inturn, ternatives. Wenowturntoaset ofcriteriathatonecan thismeansthattheoptimizationalgorithmusedtosolve usetoevaluateaproposedMDOformulationandgauge thereformulationof(1)mustshouldertheeffortofsolv- theeffectsoftheformulationonoptimizationalgorithms. ing part of theMDA problem. This is problematical if Someoftheseconsiderations,suchasdisciplinaryauton- theMDA requires specialized techniques. Moreover, if omy andper-iterationcost, are widelynoted. However, the interdisciplinarycoupling in the MDA has a domi- othercriteriaseemrarelytakenintoaccount,despitetheir nant effect, then avoiding the MDA may be inefficient. paramountimportance. Thus,insomecasesMDAmaybeunavoidable. EquivalenceofFormulations DecompositionandDisciplinaryAutonomy Ifwetake(1)asrepresentingtheMDOproblemweide- Thisisanotherveryimportantissue. Formanyotherrea- ally wish to solve, it is natural to ask whether an alter- sons (e.g., organizational lines of communication, soft- nativeformulationisequivalenttothisoriginalproblem. wareintegration),itissimplertoimplementanapproach Thereisthequestionofmathematicalequivalence: Ifa thatavoidstheiterationrequiredtosolvetheMDA.One vectorofdesignvariablessolves(1),then,suitablytrans- has a natural coarse-grained decomposition along the formed,doesityieldasolutionofthealternativeformu- lines of the disciplines; indeed, the question is not one lation,andconversely? Ifthereformulationisnotmath- ofdecompositionbutintegration. 2 Proc.,FirstASMOUK/ISSMOCONFERENCEonEngineeringDesignOptimization,July8–9,1999,MCBPress Unfortunately, in general one should expect disci- iteration, but typically theoverall cost of solvingprob- plinary autonomy to be in direct conflict with overall lemsviathisdecompositionisgreaterthanthatofsolv- computational efficiency in the optimization (see the ingtheproblemsdirectlyviathesimplexmethod,unless comments on efficiency below). Nonetheless, where theproblemexhibitsaparticularstructure. Similarcom- thecouplingbetween disciplinesis nottoogreat, disci- mentsholdforsolvingnonlinearoptimizationproblems. plinaryautonomymightnot have toodeleterious an ef- fectinthisregard. Dimensionality Some decomposition approaches, such as Collabo- Anotherquestionthatarisesisthedimensionoftheopti- rativeOptimization, make useof disciplinaryoptimiza- mizationproblemsthatensuefromagivenformulation. tioncapabilities, whichis seen bysome as an attractive Arguably, thesmaller thedimension, thebetter. For in- feature. However, the ends to which this capability is stance,theMDA(2)canbeviewedasavariablereduc- used isoftentosolvepartofthemultidisciplinaryanal- tionmethodinsofarasittreats asafunctionofthede- ysis problem. Attemptingtruemultiobjectiveoptimiza- signvariables andthusremouves from theoptimiza- tionordistributedoptimizationof aseparable objective tionproblem. x u isnotwidelydoneinMDOatpresent,andisdifficultto Ontheotherhand,anyattempttorelaxtheMDAwill accomplishforsomeserioustechnicalreasons. leadtotheintroductionofthesomeofthe intotheopti- Oneotherfeatureofdisciplinaryautonomyisitsin- mizationproblem. These additionaldegreues offreedom herentparallelism;computationcanbecarriedoutinde- inthedesignproblem arethen removed by therequire- pendently at the discipline level. However, we do not mentthattheMDAequationsbesatisfiedattheoptimal wishtoover-emphasize thisas amotivationfordecom- design. positionapproachesto(1);thesimplicityofdisciplinary An attractive feature of some decomposition meth- autonomy is its primary attraction. Moreover, the ben- ods is that one can also eliminate some of the design efitsofparallelismaresomewhat limitedbecauseofthe variables fromthesystem-leveloptimizationproblem. disparityincomputationalloadbalancing thatoftenoc- This can xbe done, for instance, if the effect of some of curs (e.g., computational fluid dynamics takes a much thedesignvariablesisrestrictedtoaspecificdiscipline. longertimethanstructuralanalysis). Oftenasequential ThisstrategyisfollowedinCollaborativeOptimization, processingofthedisciplinesmakesmoresenseforphys- aswewilldiscuss. icalandcomputationalreasons. Anotherquestionrelatedtodimensionalityisthatof thebandwidthand strengthoftheinterdisciplinarycou- WorkperIterationvs. OverallEfficiency pling. Dependingonhow (1)is formulatedand solved, the amount of information that must be exchanged be- Anostensibleattractionofapproachestosolving(1)that tween disciplines, and the frequency with which infor- arebased onreformulatingandsolvingtheproblemde- mationmustbeexchanged,mayvarymarkedly. This,in composed alongthedisciplinarylinesisthatthecostin turn,isrelatedtotheextenttowhichtheproblemisbe- eachoptimizationiterationmaybemuchlessthanthatin ingtreatedinadecomposed way, whichisitselfrelated asingleiterationof applyingan optimizationalgorithm totheefficiency withwhichtheoverallproblemwillbe directlytotheoriginalproblem(1).However,ifthecou- solved. plingbetween anyofthedisciplinesstronglyinfluences the system behavior, this may prove a false economy. TreatmentofFeasibility As a general rule in optimization, algorithms based on decomposition or separability applied to truly coupled This consideration, central to the taxonomy of MDO problemsaremuchlessefficient,overall,thanalgorithms methods,formsthesubjectofthenextsection. thatworkwiththeentirelycoupledsystem. Robustness Thisshouldnotbesurprising,asthefollowingsimple illustrationmakesclear. Supposeonewishestominimize Giventheexpenseofdesignoptimization,oneshouldde- an unconstrained, positive definite quadratic in , and mandrobustnessfromanyproposedformulationandal- supposethereare componentsof . Newton’smxethod gorithmforitssolution.Someapproachestosolvingthe costs 2 workfnorasingleiteratioxn,butfindsthesolu- MDOproblem(1)actuallyamounttosolvingsomeman- tioninOo(nnly)oneiteration. Steepestdescent,ontheother ner of relaxation or approximation of (1), but may en- hand,costsonly workforasingleiteration,but,if counterdifficultiesasonemakestherelaxationmorelike thequadratichasOhi(gnh)lyelongatedlevelsets,thensteep- theoriginalproblem. Wegiveanexampleofthisbelow, est descentwilltakefarmorethan iterationstoarrive whereaproposedformulationsuffers froma deficiency near thesolution,negatingthesmalnlerper-iterationcost thatcan defeat numerical optimizationalgorithms. One ofsteepestdescent. can fine-tune thisapproach so that for a given problem This is generally thecase for morecomplicated op- itstablyproducesanswers(thoughanswersonlytoare- timizationproblems. TheDantzig-Wolfedecomposition laxed version of the design problem); the necessity of forlinearprogrammingproblemsrequireslessworkper fine-tuningmightbeacceptableforsomesituations,but 3 Proc.,FirstASMOUK/ISSMOCONFERENCEonEngineeringDesignOptimization,July8–9,1999,MCBPress notinothers. Moregenerally, onewouldprefertohave where,given , isthesolutionoftheMDA arobustapproachfromthestart. x (u1;u2) (4) SolubilitybyAvailableAlgorithms/Convergence A1(x0;x1;u1;T1(u2)) = 0 (5) One question that is, surprisingly, sometimes over- A2(x0;x2;u2;T2(u1)) = 0: looked, is the existence of optimizationalgorithmsthat The design variables have been partitionedinto willsolveaparticularformulationoftheMDOproblem. . The sysxtem-level design variables xar=e ItispossibletoreformulatetheMDOprobleminaway (sxha0r;exd1;bxy2b)othdisciplines. Thedisciplinarydesigxn0vari- thatisdifficulttosolvereliably. Wetouchonthisinan ables and arespecifictodisciplines1and2. example below; in Alexandrov and Lewis (1999a) one Thxe1operaxt2ors and indicate that perhaps only can find adetailed illustrationof twoequivalentformu- a subset of the staRtei variaSbiles is required to evalu- lationsthatmanifestdrasticallydifferentbehaviorwhen atethesystem-levelobjective aunidthedesignconstraint conventionaloptimizersareappliedtotheirsolution. In .Theconstraints aretfhedisciplinarydesigncon- particular, formulations that lead to nonconvex bilevel gst0raints. g1;g2 and multilevel problems are hard to solve reliably, and Theoperators indicatethattheoutputofonedis- areexpensivetosolve,aswell. ciplinaryanalysismTiayneedtobetransformedbeforebe- Wemustalsoaskabouttheconvergencepropertiesof ingpassed totheotherdiscipline. Equations(4)–(5)are optimizationalgorithmsappliedtoagivenformulationof the disciplinary analysis constraints. They distinguish theMDOproblem. Attheveryleast,onewouldaskfor thedesign problem from the conventional NLP. At this aguaranteeofconvergencefromanarbitrarystartingde- stage, thereare noexplicitinterdisciplinaryconsistency signtoat least a localoptimizerof thedesign problem, constraints. sincesuchguaranteesaretypicallypartoftheanalysisof Alternativeformulationsof(3)relyontheintroduc- modernnonlinearoptimizationalgorithms.Thereisalso tion of auxiliary variables and consistency constraints. thequestionoftherateatwhichoptimizationalgorithms Forinstance,wecanrewritetheMDA(4)–(5)as willconverge,whichinpartdeterminesoverallefficiency oftheoptimization. (6) The availability of algorithms to solve a particular A1(x0;x1;u1;u12) = 0 (7) formulation of the MDO problem may limit the set of A2(x0;x2;u2;u21) = 0 (8) problemsforwhichtheformulationisuseful. Similarly, u12(cid:0)T1(u2) = 0 (9) proposedalgorithmsforthesolutionofagivenapproach u21(cid:0)T2(u1) = 0: willbelimitedbytheirconvergenceproperties. Thus we can rewrite (3) as an equivalent problem in CLASSIFICATIONOFMDOFORMULATIONS : (x0;x1;x2;u12;u21) We now turn to a classification of MDO formulations. minimize The taxonomy we propose differs from other schemes subjectto f(x0;R1(u1(x;u12));R2(u2(x;u21)) that have appeared (e.g., Cramer, et al., 1994, Balling g0(x0;S1(u1(x0;x1;u12)); andSobieski,1994). S2(u2(x0;x2;u21)))(cid:21)0 Theproposedclassificationisbasedonthewaythata g1(x0;x1;u1(x0;x1;u12))(cid:21)0 formulationhandlestheconstraintsexplicitandimplicit g2(x0;x2;u2(x0;x2;u21))(cid:21)0 in(1).Theseconstraintscomprisethefollowing: u12(cid:0)T1(u2(x0;x2;u12))=0 Disciplinaryanalysisconstraints,whichareequal- u21(cid:0)T2(u1(x0;x1;u21))=0; (10) (cid:15) ityconstraintsimplicitindisciplinaryanalyses; where,given , and arefoundbysolv- ingthediscip(lixn;aury12a;nua2ly1)sisue1quatioun2s Design constraints, which are general nonlinear (cid:15) constraints, some at the disciplinarylevel, others thatcoupleoutputsfromdifferentdisciplines; A1(x0;x1;u1;u12) = 0 Interdisciplinary consistency constraints, which A2(x0;x2;u2;u21) = 0: (cid:15) areauxiliaryconstraintsintroducedtorelaxinter- Equations (8) and (9) are examples of interdisciplinary disciplinarycoupling. consistency constraints. The degrees of freedom intro- ducedby expandingtheset ofoptimizationvariables to Wewillillustratethesedistinctionsforthefollowing include are removed by the consistency con- two-disciplineinstanceoftheMDOproblem(1): straints.u12;u21 minimize Approaches to MDO problems are generally based subjectto f(x0;R1(u1(x));R2(u2(x))) ontechniquesforeliminatingvariablesfromtransforma- (3) g0(x0;S1(u1(x));S2(u2(x)))(cid:21)0 tions of the original problem. The variables are elimi- g1(x0;x1;u1(x))(cid:21)0 nated by enforcing varioussubsets of the constraints in g2(x0;x2;u2(x))(cid:21)0; 4 Proc.,FirstASMOUK/ISSMOCONFERENCEonEngineeringDesignOptimization,July8–9,1999,MCBPress different ways. We will say that an MDO formulation formulation. Once again, closure with respect to isclosedwithrespect toagivensetofconstraintsifthe disciplinaryandsystem-leveldesignconstraintsis formulation—ratherthan an optimizationalgorithm for determined bythekindofoptimizationalgorithm itssolution—assumesthattheseconstraintsaresatisfied used. ateveryiterationoftheoptimization. Iftheformulation does not necessarily assume that a set of constraints is satisfied, we willsay thatthat formulationis open with ASTUDYOFTWOFORMULATIONS respecttothesetofconstraints. Forinstance,considerconventionaloptimizationap- We now consider members of two classes of formula- plied to the formulation(3). We perform a multidisci- tions and comment on some of their features in terms plinaryanalysisateachstep. Thiscorrespondstomain- of our previous discussion. We choose the two repre- tainingclosureofallthedisciplinaryanalysisandinter- sentativesbecausetheirapproachesareverysimilar,but disciplinaryconsistencyconstraintsin(10). aseemingly slightdifference intheproblem statements Moregenerally,MDOformulationsarecharacterized causes their analytical behavior and the effect on opti- bytheconstraintsetswithrespecttowhichtheyareopen mizationalgorithmstodiffersignificantly.Forsimplicity orclosed. Inaddition,aparticularoptimizationmethod ofexposition,thediscussionwillproceedintermsofthe appliedtotheformulationmay enforceclosurewithre- two-disciplineproblemoftheprecedingsection. specttoadditionalsetsofconstraints.However,weagain stresstheimportanceofdifferentiatingbetweentheprop- CollaborativeOptimization ertiesofaformulationandthepropertiesofanalgorithm foritssolution. Inourclassificationscheme,CollaborativeOptimization Wedistinguishthefollowingclassesofformulations, (CO) (Braun, 1996, Braun et al., 1997) is closed with based on their treatment of constraints. We refer the respect to disciplinary analyses, closed with respect to readertoAlexandrovandLewis(1999b)fordetails. design constraints, and open with respect to interdisci- plinaryconsistencyconstraints.COhasthreesalientfea- CDA/OD/CIC:Closeddisciplinaryanalysis, open tures. (cid:15) design constraints, closed interdisciplinary con- sistency constraints. This is the conventional First,COisanonconvex,nonlinearbileveloptimiza- formulation (3), also known as the Multidisci- tionproblemofaspecialstructure. plinaryFeasible (MDF)formulationinCramer et Second, the only constraints of the system-level al. (1994). Further closurewithrespect to disci- problemaretheinterdisciplinaryconsistencyconstraints plinaryand system-level design constraintsisde- thataredesignedtodrivethediscrepancyamongthedis- termined by the kind of optimization algorithm ciplinaryinputs and outputsto zero. The values of the used. Anotherexampleofaformulationincluded system-level constraintsare computed bysolvingdisci- in this large class is Bi-Level Integrated System plinaryoptimizationproblems. The number ofthecon- Synthesis(BLISS) bySobieszczanski-Sobieski et sistency constraints is related to thenumber of the dis- al.(1998). ciplines,the numberof variablesshared among thedis- ciplines, and the number of outputs exchanged among CDA/CD/OIC: Closed disciplinary analysis, thedisciplines. The formoftheconsistency constraints (cid:15) closed design constraints, open interdisciplinary characterizesdifferentinstancesofCO. consistency constraints. Examples of this class includeCollaborativeOptimization(Braun,1996, Finally, the disciplinary problems are NLP whose Braunet al., 1997)and theformulationproposed objective is to minimize the discrepancy between the inWalshetal.(1992). system-level variables and their local, disciplinary copies, subjecttosatisfyingthedesignconstraints. The CDA/OD/OIC:Closeddisciplinaryanalysis,open disciplinaryconstraints do not depend explicitly on the (cid:15) design constraints, open interdisciplinary consis- system-levelvariablesthatarepasseddowntothedisci- tency constraints. The IndividualDisciplineFea- plinaryproblemsasparameters. sible (IDF) approaches discussed in Cramer et al. (1994)and Lewis (1997)are examples of this Reformulating(10)alongthelinesofCO, weintro- class. Again, closure withrespect todisciplinary ducenewdisciplinarydesignvariables 1 2 thatserve andsystem-level designconstraintsis determined to further relax the coupling betweenx0th;ex0disciplines bythekindofoptimizationalgorithmused. throughtheshared system-leveldesignvariables . At thesystemlevel,weintroducenewvariables x0and OA/OD/OIC: Open analysis, open design con- torelax thecouplingbetweendisciplinwe1s;twhr2ough (cid:15) straints, open interdisciplinary consistency con- yth1e;ys2ystem-level objective and constraint , respec- straints. Simultaneous Analysis and Design tively. Thesystem-level cofnstraint willbeg0treatedas (Haftka,etal.,1990)isanexampleofthisclassof anadditionaldiscipline.Theresultingg0system-levelNLP 5 Proc.,FirstASMOUK/ISSMOCONFERENCEonEngineeringDesignOptimization,July8–9,1999,MCBPress in is Kroo(1996)). Inthisformulation,theconsistencycon- (x0;u12;u21;w1;w2;y1;y2) ditionistodrivetozerotheminimumvalueofsubprob- lems(12)–(14).Thesystem-levelconsistencyconstraints ms.ti.n f(x0;R1(u1(x00;x12;y1));R2(u2(x0;x21;y2))) aresimplytheoptimalvalues oftheobjectives in(12)– c0(x0;y1;y2;x0(x0;y1;y2); (14).Bearinginmindthat z1(x0;y1;y2);z2(x0;y1;y2))=0; 1 c1(x0;u12;y1;x0(x0;x12;y1); 0 0 x1(x0;x12;y1);u1(x0;x12;y1))=0 x0 = x0(x0;y1;y2) 2 1 1 c2(x0;u21;y2;x0(x0;x21;y2); x0 = x0(x0;x12;y1) 2 2 x2(x0;x21;y2);u2(x0;x21;y2)=0 (11) x0 = x0(x0;x21;y2) (15) 1 where the are the consistency constraints we u1 = u1(x0(x0;x12;y1);x1(x0;x12;y1)) 2 will shortlycdiescribe. We compute , u2 = u2(x0(x0;x21;y2);x1(x0;x21;y2)) 1 , and by xso0l(vxin0;gxt1h2e;yfo1)l- z1 = z1(x0;y1;y2) xlo1w(xin0g;xm12in;yim1)izationup1r(oxb0l;exm12i;ny1) at the disci- z2 = z2(x0;y1;y2); 1 plinelevel: (x0;x1) wehavetheconsistencyconstraints minimize 1 2 c0(x0;y1;y2)= +(cid:13)(cid:13)+x(cid:13)(cid:13)0ST(cid:0)11((xuu011(((cid:13)(cid:13)xx1010;;xx11))))(cid:0)(cid:0)uy112(cid:13)(cid:13)22 (12) c1(cid:13)(cid:13)(xx000;(cid:0)x1x20;y(cid:13)(cid:13)12)+=kzx110(cid:0)(cid:0)yx10k22+kz2(cid:0)y2 k2 (16) subjectto g1(cid:13)(cid:13)(x10;x1;u(x10;x1))(cid:21)0:(cid:13)(cid:13) +kS1(u1)(cid:0)y1(cid:13)(cid:13)k2+kT1((cid:13)(cid:13)u1)(cid:0)u12k2 2 2 An analogous problem for discipline 2 defines c2(x0;x21;y2)= x0(cid:0)x0 2 , ,and : +kS2(u2)(cid:0)y2(cid:13)(cid:13)k2+kT2((cid:13)(cid:13)u2)(cid:0)u21k2: x0(x0;x21;y2) x2(x0;x21;y2) u2(x0;x21;y2) WecallthisversionCO ,wherethesubscript2refersto minimize 2 2 thefactthatthe aresu2msofsquares. (cid:13)(cid:13)+x0S(cid:0)2(xu02((cid:13)(cid:13)x20;x2))(cid:0)y2 2 (13) tionA(1n6a)ltiesrtnoaetixvcpeilitcoittlhyemsyastctehmth-leevsyelstceomn-sliesvteenlcvyarcioabnldeis- (cid:13) 2 (cid:13) 2 +(cid:13)T2(u2(x0;x2))(cid:0)u21(cid:13) withtheirsubsystemcounterpartscomputedinsubprob- subjectto (cid:13) 2 2 (cid:13) g2(cid:13)(x0;x2;u(x0;x2))(cid:21)0:(cid:13) lems(12)–(14): Inthedisciplinaryproblems, and arecomputedvia thedisciplinaryanalyses u1 u2 0 c0(x0;y1;y2)=(x0(cid:0)x0;z1(cid:0)y1;z2(cid:0)y2) 0 c1(x10;u12;y1)= (17) A1(x0;x1;u1;u12) = 0 (x0(cid:0)x0;S1(u1)(cid:0)y1;T1(u1)(cid:0)u12) 1 A2(x0;x2;u2;u21) = 0: c2(x0;u21;y2)= 2 “Discipline0”isintroducedtotreatthesystem-levelde- (x0(cid:0)x0;S1(u2)(cid:0)y2;T1(u2)(cid:0)u21); sign constraints; the associated disciplinaryproblem in again, keeping in mind (15). We will denote this ap- is proach as CO . In general, this leads to more system- 0 (x0;z1;z2) levelequality1constraintsthandoesCO . Thelatterusu- minimize 0 2 2 ally reduces this vector of information2about inconsis- x0(cid:0)x0 +kz1(cid:0)y1 k (14) tency into as many constraints as there are subsystems (cid:13) (cid:13) 2 (cid:13) +kz2(cid:13)(cid:0)y2 k (Braun,1996),butthevectormaybereducedtoasingle subjectto 0 g0(x0;z1;z2)(cid:21)0: scalar. The introductionof disciplinaryminimization subprob- Note that the number of system-level variables in lems ofthisformisthedistinctivecharacteristic ofCO. thesystem-levelproblem(11)dependsonthenumberof Thesubproblemsareindependentofoneanotherandcan sharedvariables andthebandwidthoftheinterdisci- be solved autonomously at the discipline level. In do- plinarycoupling,xa0smanifestin . ingso,thedisciplinarydesignvariables andthedisci- Theintroductionoftheseauxiliaruy1v2;arui2a1b;lews1s;uwg2g;eys1t;thya2t plinarystatevariables areeliminatedfxroimthesystem- COwillbebest suitedforproblemswithanarrowcou- levelproblem. ui plingbandwidth. COpossesses amarkeddegreeofdis- Information from the solutions of the disciplinary ciplinary autonomy. However, as computational expe- problems (12)–(14) is then used to define the system- rience (Alexandrov and Kodiyalam, 1998, Kodiyalam, level consistency constraints . Here we will discuss 1998) and analysis (Alexandrov and Lewis, 1999a) re- twodefinitionsof . ci veal, the approach has a number of intrinsic analytical The first instancice of CO we discuss is the form in andcomputationaldifficulties. which CO is usually presented (Balling and Wilkinson For instance, in CO , one can show that Lagrange (1997),Braun(1996),Braunetal. (1997),Sobieskiand multipliers never exist f2or the system-level constraints 6 Proc.,FirstASMOUK/ISSMOCONFERENCEonEngineeringDesignOptimization,July8–9,1999,MCBPress (16). Thismeansthatnonlinearoptimizationalgorithms straints;i.e., and aredefinedbysolving willfailifoneattemptstotrulyenforcetheconsistency u1 u2 conditions ; instead, one must be content witha relaxation ci = 0 forsomesuitablysmall . Inprac- A1(x0;x1;u1;u12) = 0 tice, the lakrgceirk(cid:20)at"the system level and the"tighter the A2(x0;x2;u2;u21) = 0: convergencecrit"eriaforthesubsystems,thebetterarethe Incontrast to CO, theIDF formulationis a single-level chances that an optimizationalgorithm applied to CO NLP. willfindasolution. 2 In the IDF approach, further closure with respect This problem does not occur in CO . However, in to disciplinary design constraints or system level con- ordertocomputethefirstderivativesint1hesystem-level straints is determined by the kind of optimization al- optimizationproblem we must compute second deriva- gorithm used. Ideally, one would be able to start with tives at the disciplinary level; this is a consequence of design variables for which the disciplinary the fact that CO involves a bilevel optimization prob- design constraint(sxd0e;fixn1e;dx2b)y the are satisfied. One lem,inwhichthesystem-leveloptimizationproblemin- could then apply an optimizationgailgorithm that main- volves the output of the disciplinary level optimization tainedfeasibilitywithrespecttotheseconstraintssothat problems. allsubsequentdesignsobtainedinthecourseoftheop- Other difficultiesinbothCO and CO derivefrom timization satisfied the disciplinary design constraints, the fact that CO leads to a nonl1inear bile2vel optimiza- thereby accomplishing the same end that CO achieves tionproblem. Inparticular,theconstraints(16)and(17) through the definition of its disciplinary optimization possess features thatcan cause distressforstandardop- problems. timization algorithms. These include the potential for On the other hand, one might rightly object that it nonsmoothnessduetomultiplelocalminimainthedis- will,ingeneral,bedifficulttofindinitialdesignvariables ciplinary problems (12)–(14). For further details, see forwhichthedisciplinarydesignconstraints AlexandrovandLewis(1999a). (axre0;saxt1is;fixe2d). Toaddressthisproblem,wecanexpandthe spacealongthelinesofCOasfollows: IndividualDisciplineFeasibleApproaches TheIDF formulationprovidesanotherway toavoidthe minimize expensive MDA iteration. It is closed with respect to subjectto f(x0;R1(u1);R2(u2)) 0 disciplinary analyses, open with respect to design con- g0(x0;y1;y2)(cid:21)0 1 straints, and open with respect to interdisciplinarycon- g1(x0;x1;u1)(cid:21)0 2 sistency constraints. Furtherclosurewithrespect tode- g2(x0;x2;u2)(cid:21)0 signconstraintsorsystemlevelconstraintsisdetermined u12(cid:0)T1(u2)=0 (19) bythekindofoptimizationalgorithmused. u21(cid:0)T2(u1)=0 Various forms of IDF were discussed in Cramer et 0 x0(cid:0)x0 =0 al. (1994)andLewis(1997). Theterm“IndividualDis- 1 x0(cid:0)x0 =0 ciplineFeasible” originallyreferred tomaintainingclo- 2 x0(cid:0)x0 =0 surewithrespect tothedisciplinaryanalysisconstraints at each optimization iteration, but not closure with re- y1(cid:0)S1(u1)=0 spect to multidisciplinary analysis coupling until a so- y2(cid:0)S2(u2)=0: lutionisreached. However, thequestionofdisciplinary Thisrelaxes therequirementthatthedisciplinarydesign designconstraintswasnotreallytreatedinearlierdiscus- constraints be satisfied with the system-level values of sionofIDF.HerewediscusstheoriginalIDFapproach . In particular, we now have the flexibility to select and its elaborationthat treats thedesign constraintsex- xth0einitial i and inawaythatthedisciplinarydesign plicitly. constraintsxa0re satyisified, exactlyas inCO.Onecan then Webeginwith(10). TheIDFformulationdiscussed applyanoptimizationalgorithmthatenforcesfeasibility in(Cramer et al., 1994, Lewis, 1997)isclosed withre- withrespecttothedisciplinarydesignconstraints. specttothedisciplinaryanalysisconstraints: ItisstraightforwardtoverifythatIDFisequivalentto theoriginalMDO.Thismakes IDFeasy toanalyze; for minimize instance,ifstandardconstraintqualificationsaresatisfied f(x0;R1(u1);R2(u2)) subjectto bytheoriginalproblem,thentheyalsoholdfortheIDF g0(x0;S1(u1);S2(u2))(cid:21)0 formulation.Theconvergencepropertiesofoptimization (18) g1(x0;x1;u1)(cid:21)0 algorithmsappliedtoIDFarethoseofthealgorithmsap- g2(x0;x2;u2)(cid:21)0 pliedtoconventionalNLP.Givenagoodsolverforequal- u12(cid:0)T1(u2)=0 ityconstrainedoptimizationproblems,themethodisex- u21(cid:0)T2(u1)=0; pectedtobeefficient. where and SimilarlytoCO, IDF is intendedforproblems with arerequui1re=dounl1y(xto0;sxat1i;sufy12th)ediscuip2li=naruy2a(nxa0l;yxs2is;uco2n1)- small bandwidth of interdisciplinary coupling, and the 7 Proc.,FirstASMOUK/ISSMOCONFERENCEonEngineeringDesignOptimization,July8–9,1999,MCBPress problem of decomposition is similar to that of CO. MultidisciplinaryDesignOptimization:StateoftheArt, Also similarly to CO, formulationsthat arise from IDF SIAM,Philadelphia. havemoreoptimizationvariablesthatthosearisingfrom Alexandrov,N.M.andKodiyalam,S.(1998),“Initialre- MDF. sultsofanMDOmethodevaluationstudy”,AIAAPaper Importantly,althoughIDFmaintainsautonomywith 98-4884. respecttoanalyses,itlacksCO’sautonomywithrespect to disciplinary optimization. That is, while the anal- Alexandrov,N. M.andLewis, R.M.(1999a), “Analyti- calandcomputationalaspectsofcollaborativeoptimiza- yses are performed autonomously during the analysis tion”,tobesubmittedtoEngineeringOptimization. stage, the coupling is restored during the optimization stepcomputation.Thisbringsbackthedifficultiesofin- Alexandrov, N. M. and Lewis, R. M. (1999b), “Prob- tegration.Ontheotherhand,aspreviouslynoted,thedis- lemformulationandalgorithmsinmultidisciplinaryop- ciplinaryoptimizationinCOisactuallypartofaddress- timization”,ICASEreport. Inpreparation. ingtheMDA,notactuallyimprovingthedisciplinaryob- Balling, R. J. and Sobieszczanski-Sobieski S. (1994), jectivesofthetruedesignproblem,suchasweightorthe “Optimization of coupled systems: a critical overview lift-to-dragratio. ofapproaches”. AIAAPaper94-4330. CONCLUDINGREMARKS Balling, R. J. and Wilkinson, C. A. (1997), “Execution of multidisciplinarydesign optimizationapproaches on This paper has introduced a portionof an extensive ef- commontestproblems”,AIAAJ.,35,pp.178–186. fortaimedatfurtheringtheunderstandingoftheanalyti- calandcomputationalpropertiesofmethodsforsolving Braun, R. (1996), Collaborative Optimization: An ar- MDOproblemsandatproposingefficientmethodsbased chitectureforlarge-scaledistributeddesign,PhDthesis, on this understanding. We emphasized the distinction StanfordUniversity. betweenanMDOformulationandanoptimizationalgo- Braun, R. D., Moore, A. A. and Kroo, I. M. (1997), rithmanddiscussedanew,comprehensiveclassification “Collaborativeapproachtolaunchvehicledesign”,J. of of MDO formulations based on the way the constraint SpacecraftandRockets,34,pp.478–486. sets areexplicitlytreatedinaformulation. Members of twoformulationclasseswerediscussedasanexample. Cramer, E., Dennis, Jr., J. E., Frank, P., Lewis, R. and WhenconsideringanMDOproblemstatement,both Shubin,G.(1994),“Problemformulationformultidisci- theproblemformulationandtheavailablenonlinearpro- plinarydesign optimization”,SIAM J. on Optimization, gramming algorithms for solving the formulation must 4,pp.754–776. be examined carefully with the following questions in Haftka, R. T., Gu¨rdal, Z. and Kamat, M. P. (1990),El- mind: How is the new formulation related to the ba- ements of Structural Optimization, Kluwer Academic sic NLP formulation of the MDO problem? Does the Publishers,Dordrecht. newformulationleadtoanoptimizationproblemthatis Lewis,R.M.(1997),”PracticalAspectsofVariableRe- notamenable to solutionbyexistingoptimizationalgo- ductionFormulationsand ReducedBasis Algorithmsin rithms?Forinstance,isthenewformulationanonconvex Multidisciplinary Design Optimization”, in Multidisci- multileveloptimizationproblem? plinaryDesign Optimization: State-of-the-Art, Alexan- Themotivationforaprospectiveformulationmustbe drov,N.M.andHussaini,M.Y.,Eds.,SIAM. continuallyre-examined. Itmaybediscoveredthatwhat oneattemptstoaccomplishviaaformulationwithadiffi- Sobieski, I. and Kroo, I. (1996), “Aircraft design using cultstructureismoreeasilyaccomplishedbyajudicious collaborativeoptimization”. AIAAPaper96-0715. choiceofanoptimizationalgorithm. Sobieszczanski-Sobieski, J., Agte, J. and Sandusky, Asarule,fulldisciplinaryautonomywithrespectto Jr., R. 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