The state of the art in Collaborative Design
Mohamed Masmoudi
y
Didier Auroux z
Yogesh Parte
Abstract
ThegoalofthispaperistopresentanddiscussmaincontributionsinMultiDisciplinaryOptimiza-
tion(MDO) orcollaborativedesign. Thegoalof MDOisto deneamethodology fordisciplines
interactionwiththeaimofsolvingaglobaldesignproblem. InMDO,mostoftheauthorsconsider
twotypesofparameters,public(sharedbydisciplines)parameters,andprivateones. Inthispaper,
werstfocusourinterestonthedenition,creationanduseofpublicparameters. Wegivethenan
overview ofmostclassicalMDOapproaches. Finally, wepresentanother wayto dealwith MDO
problems,basedongametheory. Variousexamplesaregivenin thefollowing.
Key Words: Multidisciplinaryoptimization,collaborativedesign,gametheory.
1 Introduction
The goal of this paper is to present and discuss
most signicant contributions in M ulti-Disciplinary
Optimization(MDO).
The name multidisciplinary optimization is often
takenliterally. Wepreferto call itcollaborativede-
sign. Indeed, theoptimization toolis onlyone part
whichcannotbeseparatedfromthetotaldesignpro-
cess. Moreover,thegoalisnottocreateanautomatic
designprocessbasedonoptimizationalgorithmsbut
to alloweasyinteraction betweenteams fromdier-
entdisciplines.
Thecommon pointbetweenthedierent publica-
tions is to consider twotypes of design parameters,
publicparameters(sharedbydisciplines)andprivate
parameters(specic to thegiven discipline). Unlike
mostcontributions,weconsideronlypublicparame-
tersinthispaper. Weassumethatforagivenchoice
ofpublicparameters,privateparametersarexedby
eachdisciplineat theiroptimalvalue.
Thedenitionofpublicparametersandtheirrange
of validityare big issuesin MDO.Generally, the -
nal design depends strongly on the set of valid pa-
rameters. Forexample, for the design of a wing, a
thicknessandthehopeofCFD(ComputationalFluid
D ynamics) team isto decrease it. Parameterization
is awayto help teams to nd acompromise. Only
fewpaperspointouttheroleofparametricdenition.
ShapeparametersarequitediÆculttohandle. En-
gineers usually create parameters on dead meshes.
Unfortunately, eachdisciplinehas itsown mesh: in-
side thestructurefor structuralanalysis,on itssur-
face for acoustic, and outsideit for CFD. For these
reasons,wehavetodenedisciplineindependentpa-
rameterizedshapes.
Nowadays standard CAD (Computer Aided
D esign)codesprovideparameterizationfacilitiestak-
ing into account bounds and other types of con-
straints. ShapedenitionbyCADisanexcellentway
tofacilitateinteractionbetweendesignteams. Unfor-
tunately, this is notthe case for engineering teams.
CADcodesdonotoermeshparameterizationfacil-
ities. UsingCAD environment,itisonlypossibleto
buildanewmeshforagivenchoiceofdesignparam-
eters. Thiswayofgenerating meshesisnotsuitable
for optimal shape design. Generally, the numerical
error due to the change in topology of the mesh is
toohighcomparedtotheinformationtobecalculat-
ed(thedierencebetweenverycloseresponses).
ingaplatformallowingtheinteractionbetweenanal-
ysissoftwaresandoptimizationtoolboxes. Thewide-
ly known softwares are iSight, Dakota, Optimus
(from LMS)and Boss Quatro[64, 65,58, 2]. These
environmentsalsosupport \blackbox" applications.
The language PEARL for le parsing is commonly
used.
Considering these points, the paper is organized
as follows. In section 2, we discuss the parameter
denition problem. Most classicalMDOapproaches
arepresentedin section 3. Theapplication of game
theoryisarecentcontributionofJ.Periauxetal. to
MDO. It will be presented in section 4. In section
5wegiveabriefsurveyof veryrecentdevelopments
andalternativeapproachesforMDO.
2 Denition of parameters
2.1 Public parameters
In a multidisciplinary context, the choice of the
parameters can lead to many negotiations between
teams. Thedenitionorchoiceofparametersisitself
adesign step, and the nal resultstrongly depends
onit.
In our opinion, the denition of parameters is a
majorissueinthecollaborativedesign. Despitetech-
nological developments, this problem remains diÆ-
cult. It cannot be dened in a design department
(e.g. CAD department). Itis theaairof engineer-
ing department; and generally the rst simulations
carried out on the product allow parameter deni-
tioninarelevantway.
yLaboratoire MIP, Universite Paul Sabatier
Toulouse 3, 31062 Toulouse Cedex 9, France -
masmoudi@mip.ups-tlse.fr
zLaboratoireMIP, UniversitePaul Sabatier Toulouse 3,
31062ToulouseCedex9,France-auroux@mip.ups-tlse.fr
LaboratoireMIP,UniversitePaulSabatierToulouse3,
31062ToulouseCedex9,France-parte@mip.ups-tlse.fr
c
MohamedMasmoudiDidierAuroux&YogeshParte
SAROD-2005
Publishedin2005byTataMcGraw-Hill
2.2 Private parameters
Ifweconsiderawing,thenumberofribs,theirposi-
tionsandtheirsizesmayonlyinteresttheengineerin
chargeofstructuralanalysis. Onecallsthemprivate
parameters.
Foragivenchoiceofpublicparameters,thestruc-
ture engineering department should carry out com-
putations with an optimal conguration of private
parameters. Thisideaisconsideredinmostpapers.
2.3 Denition of public parameters
Ifoneconsidersthesignicantcaseofshapeoptimiza-
tion,thenumberofparametersareinnite. Inprac-
tice,onecanworkwithalargenumberofparameters:
allthenodesof themesh aredesignparameters. In
this case, it is possible to nd the optimal solution
(of the optimization problem), but not the optimal
design. The designcriterialikeaestheticand manu-
facturabilityarediÆculttodescribeandtheyarenot
takenintoaccountbytheoptimizationproblem. By
reducingthenumberofdesignparameterswereduce
theprobabilityofunacceptablesolutions.
However,anoptimizationprocessinvolvingalarge
numberofparameterscanhaveatleasttwoapplica-
tions:
The optimization process may compute an un-
expected form and one might realize afterward
that thecomputedform isfarfrom beingunin-
teresting,despitesomeunsatisedcriteria.
Inthedevelopmentstep,thecomputationofthe
sensitivitywithrespecttoalldegreesoffreedom
oftheproblem(usingadjointmethods)makesit
possibletondthemostsensitiveregionsandto
createmoreappropriateparameters.
It isclearthat in amultidisciplinarycontext, one
canonlywork withalimitednumberofparameters.
2.3.1 Adjoint methods
Adjoint methods are an excellent wayto dene pa-
rameters. Thecriteria ofthevarious disciplinescan
becombinedin onlyonecriterionin orderto havea
on the surface of the initial design indicates more
sensitivepartsoftheshape. Onecanalsocomputea
gradientforeachcriterion. Eachgradient,denedon
thesurface,canbeconsideredasabasicperturbation
function.
In [72] the authors determine the signicant pa-
rametersbyconsidering anhierarchicalapproach. A
methodbasedoncontrolvolumesispresentedinthe
nextsection.
Forthedenitionofadjoint,wereferto[16,50,32].
Automaticdierentiationmethodscouldbeusedfor
adjointcodegeneration[24,23,22,50,55,56,63,70,
71,78].
2.3.2 Engineeringknowledge
Thesecondapproachis basedonengineeringknowl-
edgeaboutthedimensioningparametersofashape.
This is, for example, thecase of the socalled aero-
functions likefunctions controlling the curvature of
theleadingedgeofawing.
These twoapproaches(adjointmethods and engi-
neeringknowledge)arecomplementary.
2.4 Creation of parameters
2.4.1 Overview
Inthe90's,commercial CADsoftwaresproposedfa-
cilitiesallowingtheparameterizationofshapesunder
someconstraints. These constraintscan directlyap-
pear as mathematical equations connecting one set
ofparameterstoothers(imposedsurfaceorvolume).
Theycanalsobeof geometricalnature: orthogonal-
ity,parallelism,etc.
Thesenewtoolsarestillnotused,inspiteofincen-
tive policy of many companies for the use of CAD
parameterizationfacilities. Thereasonbeing
TheyarediÆculttoimplement: Itisnotsoeasy
toimplementsimultaneouslyaninnitenumber
ofshapeswhereasitis alreadydiÆcultto draw
axed geometry.
The technical communications often rely on
CAD within a company. However, there is no
standardformatallowingparameterizedgeome-
trytransferbetweendierentsoftware.
Other concerns appear in the particular context
ofcollaborativedesign. Therelevantparametersare
notaprioriknownandCADparametersmaynotbe
suitable.
Engineersgenerallydon'tusetheCADfacilitiesfor
thecreationofparameters. Theyusedtomakesome
perturbationsofthedeadmesh.
2.4.2 Controlvolumes
We propose in this section an idea that meets the
needsofengineers. Inwhatfollows,wewillworkon
adeadmesh.
Wepresentamethod based onthevolume trans-
formations. Itconsistsinassociatingaparameterized
volume to the mesh. The mesh nodes undergo the
sameperturbationsasthepointsof thevolume(see
paragraphMeshperturbation).
ThebivariateNURBS (Non Uniform RationalB-
Splines) are a standard way for surface representa-
tion. One canimagine NURBS depending on three
variables(u;v;w)forthedenition ofvolumes. The
perturbation of the control points of the NURBS
leadsto anaturalmeshperturbation.
OnecanusehexahedralniteelementsasNURBS
of degree 1 or 2. These elements are generally im-
plemented in the computational part of most CAD
software.
Themajoradvantageofthesemethodsistocreate
aregularmeshperturbationinside eachvolume.
Mesh perturbation Theuseof NURBS ornite
elements both requirean explicitfunction,which at
eachpoint(u;v;w)ofthereferencecube(0u1,
0v 1,0w1) associatesapoint (x;y;z)of
thecontrolvolume(seeFig. 1),where
0
@
x(u;v;w)
y(u;v;w)
z(u;v;w) 1
A
= X
P
i
(u;v;w) 0
@ x
i
y
i
z
i 1
A
(1)
and where (x
i
;y
i
;z
i
) are the control points of the
controlvolume.
How tobuild aperturbation ofthemesh? Let us
denote by (x 0
i
;y 0
i
;z 0
i
) the coordinates of the control
points before the perturbation, and (x 1
;y 1
;z 1
) the
coordinates of these points after the perturbation.
Let(x 0
;y 0
;z 0
) beamesh node before perturbation.
Werstcomputethetriplet(u 0
;v 0
;w 0
)such that
0
@ x
0
y 0
z 0
1
A
= X
P
i (u
0
;v 0
;w 0
) 0
@ x
0
i
y 0
i
z 0
i 1
A
: (2)
Thecoordinatesofthispointafterperturbationare
givenby
0
@ x
1
y 1
z 1
1
A
= X
P
i (u
0
;v 0
;w 0
) 0
@ x
1
i
y 1
i
z 1
i 1
A
: (3)
Oneshouldsolvethesystemofthethreeequations
(2)andndthethreeunknownsateachnodeofthe
initialmesh. Thisoperationshouldbedoneonlyonce
forallperturbations.
Perturbation of the external mesh In the
acoustic or electromagnetic eld analysis, only the
structure's surfaceis meshed. In this case, thepro-
cess described in the previous paragraph makes it
possible to perturb the surface nodes which are by
denitioninside thecontrolvolume.
Aerodynamicphenomenaareoftenstudiedoutside
the structure. Inthis case, the greatest partof the
meshisoutsidethecontrolvolume.
We propose to use the control volume technique
forthecomputationoftheperturbationsofthemesh
surface,andtousetraditionalmethodsfor thecom-
putationofthevolumeperturbations. Therearesev-
eralmethodsofmesh propagation.
2.4.3 Mesh propagation
Inthe caseof an externalmesh, ifthe parameteris
denedbyasurfaceperturbation,oneneedstoprop-
agate the perturbation of the surface to the neigh-
bouringmesh.
Elasticitymethods Theedgeperturbationiscon-
sidered asan imposed displacement. Insidethe do-
main,onesolvesalinearelasticityproblem. Ingen-
eral,stinessisanincreasingfunctionoftheelement
volume. We use the current mesh as asupport for
Integral methods Letusdenote byF(x)theim-
ageofapointx. WeassumethatF(x)isknownfor
allpointsx oftheedge. The displacement ofanin-
ternalpointisweightedbytheinverseofitsdistance
fromthesurface:
F(x)= Z
@
r
0 F(x
0
)
kx x 0
k dx
0
!
= Z
@
r
0 1
kx x 0
k dx
0
!
:
(4)
Thisintegralmaybecomputedbyusingthenite
elementsmesh.
2.5 Application to a wing
We now consider a wing whose parametersare the
thickness,thetwist,thearrow(seeFig. 2),thebend,
andtheinection. (seeFig. 6and7).
In this case, we will naturally use the methods
basedoncontrolvolumes.
A thicknessmodicationsimplyconsists in aver-
ticalperturbationofthecontrolpointsoffaceA(see
Fig. 1and3). Thetwistconsists inarigidrotation
offace B(seeFig. 1and4). Thearrowmodication
consists in arigid translation of face B (see Fig. 1
and5).
Naturally, onecanimagineanite elementof de-
gree2ifwehaveto modify thebend of theaerofoil
ortheinectionofthewing. Onecanseein gure6
anite element of order two in the direction corre-
sponding to thebend, and in gure7anothernite
element of order twoin thedirection corresponding
totheinection.
One can see on gure 8 the engine displacement
along the wing. Blocks B1 and B3 follow the dis-
placementinducedbythecontrolpointsofblockB2.
Iftheleadingedgeandthetailingedgewereparallel,
one could only consider rigid displacements of this
block. Ifwewanttotakeinto accounttheanglebe-
tweenthese twoedges, wehave to force thecontrol
Figure1: Referencecube(left)andinitialhexahedral
elementwithits8controlpoints(right).
Figure 2: Wingparameters.
Figure3: Modicationofthewingthicknessusing a
niteelementofdegreeonein each direction.
Figure 4: Modication of the aerofoil twist using a
niteelementofdegreeonein eachdirection.
Figure 5: Modication of the arrow using a nite
elementofdegreeonein eachdirection.
Figure 6: Modicationofthebend using anite el-
ementofsecondorderin onedirectionandofdegree
onein thetwootherdirections.
Figure7: Modicationoftheinectionusinganite
elementofsecondorderinonedirectionandofdegree
onein thetwootherdirections.
Figure 8: Engine displacement. The arrows cor-
respond to the degrees of freedom of the dierent
points,theotherpointsarexed.
3 Several collaborative design
methods
Inorder to simplify this presentation, we limitour-
selvesto twodisciplines and theprivateparameters
are hidden by each discipline. The state equations
aregiveninanexplicitform
a
1
= A
1 (x;a
2
) (5)
a
2
= A
2 (x;a
1
) (6)
wherexrepresentspublicdesignparameters. Weas-
sume that one forgets the state, and we only take
careabouttheoutput. Wedenote bya
i
theoutput
of discipline i. In the rst equation, a
2
(resp. a
1 )
correspondstothecontributionofdiscipline2(resp.
1)in thecomputationofa
1
(resp. a
2 ).
Wewanttosolvethefollowingoptimizationprob-
lem
min
x
f(x;a
1 (x);a
2 (x)):
g(x;a
1 (x);a
2
(x))0
(7)
In this presentation we do notconsider disciplinary
internal constraints; theyshould be implicitlytaken
intoaccountbythestateequations.
3.1 The MDA (Multi-Disciplinary
Analysis) methods
In each iteration of theoptimization algorithm, the
stateequationsareassumedtobesatised[3]. Thisis
themostnaturalmethodinwhicheachdisciplinepro-
vides someanalysis services and possibly somegra-
dientcomputations.
3.2 The AAO (All-At-Once) method
The AAO method is also the so-called \one shot",
SAND orSAD (S imultaneous Analysis and Design)
method [15, 4, 30]. One solves simultaneously the
stateequationsandtheoptimizationproblem. Con-
trary to the MDA method, the state equations are
notsupposed tobesatisedat eachstepoftheopti-
mizationalgorithm,but theyare satisedwhen the
convergenceisachieved. Thestateequationsarein-
This method is really eÆcient, but it can be
diÆcult to obtainitsconvergence whensomeof the
state equations are strongly nonlinear. Even in a
monodisciplinary context, theconvergenceisnotal-
waysachieved.
3.3 The DAO (Disciplinary Analysis
Optimization) method
In the DAO method [2, 4], the optimization is no
more performed by the disciplines. However, each
disciplinemaychangetheparametersinordertoob-
tainanadmissiblepoint. Wepreviouslyassumedthat
theprivateconstraintsarepartofthestateequations.
Theprivate parametersare generallyused tosatisfy
theseconstraints,buttheremaynotexistanyvector
x, representing thepublic variables, i.e. solution of
theproblem. Thisiswhyoneusuallyrelaxesx. Each
disciplinechoosesitsownz
i
inordertosatisfyitspri-
vateconstraints. Whentheconvergenceis achieved,
thethree z
i
areequalif weadd three constraints in
theoptimizationproblem:
min
x f
DAO (x;b
1
;b
2 )
g(z
0
;b
1
;b
2 )0
b
1
=a
1 (z
1
;b
2 );
b
2
=a
2 (z
1
;b
1 );
z
0
=x
z
1
=x
z
2
=x
(8)
3.4 The CO (Collaborative
Optimization) method
IntheCOmethod [15, 2, 6,4,5], allthedisciplines
contributetotheimprovementoftheresults. Weuse
thesamenotationsasintheprevioussectionpresent-
ingtheDAOmethod.
Collaborative Optimization is a bilevel optimiza-
tionapproach: thesystemleveland thedisciplinary
Atthesystemlevelwesolvethefollowingproblem:
8
<
:
min
x f(x;b
1
;b
2 )
kx x
1 k
2
+kx x
2 k
2
+ka
1 b
1 k
2
+ka
2 b
2 k
2
=0
g(z
0
;b
1
;b
2 )0
(9)
The main goal of this system level optimization
problem is to impose to each disciplinesomeobjec-
tives tobereachedin a leastsquare sense. Theob-
jectivetargetisb
i
fordisciplineiandthedisciplinary
solutionx
i
shouldbeclosetox.
The disciplinary optimization problem for disci-
pline 1isthen
min
x
1 1
2
kx
1 xk
2
+kb
1 a
1 (x
1
;b
2 )k
2
(10)
wherea
1
isthesolutionofthestateequation(5).
Theoptimizationproblemfordiscipline2is
min
x
2 1
2
kx
2 xk
2
+kb
2 a
2 (x
2
;b
1 )k
2
(11)
wherea
2
isthesolutionofthestateequation(6).
3.5 The CSSO (Collaborative S ub-
S pace Optimization) method
Wewill giveaquickoverviewof theCSSOmethod,
onecansee[68]foramoredetailedpresentation. The
designparametersaredistributedbetweenthedier-
entdisciplines. Eachdisciplinehastooptimizeafew
parameters.
The CSSO is similar to the domain decomposi-
tion method and its application for solving state
equations, exceptfor oneaspect: in domaindecom-
position,anunknowncaneasilybeassociatedtoone
equation of the state. In multisciplinary optimiza-
tion,itseemshardtoassociateanypublicparameter
tooneparticulardiscipline,becauseithasbydeni-
tiontobesharedbetweendierentdisciplines.
Westillusethe samenotationsasintheprevious
sections. Fromnowon,wewilluseanapproximation
ofdiscipline2(denotedby
~
A
2
)intheequationcorre-
spondingtodiscipline1,and reciprocally,weusean
approximationofdiscipline1(denotedby
~
A
1 )inthe
Letusremindtheinitialproblem
a
1
=A
1 (x;a
2
) (12)
a
2
=A
2 (x;a
1
) (13)
min
x f(x;a
1 (x);a
2 (x))
g(x;a
1 (x);a
2
(x))0 (14)
andweset x=(x
1
;x
2
)where x
i
isthevectorofthe
variablesassociatedto disciplinei. Then, wedenote
by~a
i
theapproximationoftheanalysisofa
i
. Wewill
furthergivemoredetailsabouttheseapproximations.
Itisthennaturaltoconsiderthefollowingequations
a
1
=A
1 (x;~a
2
) (15)
~ a
2
=
~
A
2 (x;a
1
) (16)
min
x
1 f(x
1
;x
2
;a
1 (x);~a
2
(x)) (17)
g(x;a
1 (x);~a
2
(x)) 0
(1 )x
1cur x
1
(1+)x
1cur
: (18)
The last constraintis introduced to set thevalidity
domainoftheapproximationofa
2
aroundthecurrent
point x
1cur
. The approximate response ~a
2
and the
coeÆcientareprovided bydiscipline2.
Inthesameway,wecanwrite theproblemsolved
bydiscipline2
~ a
1
=
~
A
1 (x;a
2
) (19)
a
2
=A
2 (x;a~
1
) (20)
min
x
2 f(x
1
;x
2
;~a
1 (x);a
2
(x)) (21)
g(x;~a
1 (x);a
2
(x)) 0
(1 )x
2cur x
2
(1+)x
2cur
(22)
The process i is supposed to provide to all other
disciplines the state variable x
icur
, the approxima-
tion~a
i of a
i
anditstrustregion.
Many results have been published about param-
autonomyof each disciplineand enlarging thetrust
region[26,27,28,29,45,41,57,62,66, 67,69].
Inmostdomaindecompositionmethods [1],there
isanaturalwaytoassociateunknownstoequations,
andthereare asmanyequationsasunknowns. This
associationmakestheresolutionmucheasier. Incol-
laborativedesign, itis notso naturalto associatea
sub-spaceto adiscipline.
3.6 Surrogate models
Surrogatemodelsprovideaconvenientwaytoreduce
MDOcomplexity. An approximationhasalocalva-
lidityinanappropriatetrustregion:
Linearizationisvalidinasmallregionandfora
largenumberof parametersifadjointisconsid-
ered,
Response surfaces are generallyvalid in alarge
regionforonlyfewparameters.
3.6.1 First order approximation
Themostsimplewaytobuildalocalapproximation
isto dierentiate (5)and (6), but itstrustregion is
relativelysmall. Weobtain
~ a
1
=@
x A
1 (x
cur
;a
2 (x
cur
))(x x
cur )
+ @
a2 A
1 (x
cur
;a
2 (x
cur ))(a
2 (x) a
2 (x
cur ))(23)
and
~ a
2
=@
x A
2 (x
cur
;a
1 (x
cur
))(x x
cur )
+ @
a1 A
2 (x
cur
;a
1 (x
cur ))(a
1 (x) a
1 (x
cur )):(24)
The adjoint method (reverse mode in automatic
dierentiation) could be considered for large scale
problems: x
i
are of large dimension[24, 23, 22, 50,
55,56,63,70,71,78].
3.6.2 Approximation usingparameterization
methods
Parameterization methods are usually appropriate
cisely, we generallyassumethat onedisciplineis in-
dependentoftheothers:
a
1
=A
1
(x) (25)
a
2
=A
2 (x;a
1
): (26)
In this example, discipline 1 does not require any
informationfromdiscipline2.
One can see this kind of approximations in the
eld of aeroelasticity. The output a
1
usually repre-
sentsthestructureresponsetomechanicalexcitation
(a modal analysisis oftennecessaryfordetermining
thestructureresponse). Agoodapproximation~a
1 of
a
1
consists in using the samemodal basis whenthe
designparametersx vary.
Tobuildsurrogate(orreduced)models,weproceed
in the followingway. Firstwesolve(5) and (6) for
several vectors x. Generally, weconsider Design of
Experiments (DOE) methods to select appropriate
vectorsx k
,k=1;;nwherenissmall.
Thesecondstepconsistsinbuildingthesurrogate
model. There aretwofamiliesofmethods:
1. Approximationofcriteriaandconstraintsusing
surfaceresponse methods [18, 75, 60] like poly-
nomial approximation, neural networks, SVM
(Support Vector Machine), RBF (Radial Basis
Functions),...
2. Foreach vectorx k
, weconsiderthecorrespond-
ing solutionu k
(the resultof the full analysis).
Then, for a given new vector x, we look for
the corresponding solution u as a linear com-
bination of vectors u k
. This problem is smal-
l and has only a n-dimensional unknown vec-
tor. From the numerical point of view, it is
necessary to build an orthonormal basis from
vectors u k
. For this reason, this method is
calledPOD(ProperOrthogonalDecomposition)
[26,27,28,29,45, 41,57, 62,66,67,69].
Ifwecomparemethods1and2,theparameterization
providedby POD takesinto accountthe knowledge
ofthemodel.
With parameterization, theevaluation of thecost
functionissocheapthatitispossibletoconsiderge-
neticalgorithms[46, 59],and otherglobal optimiza-
4 Game theory
Game theory has been used for multicriteria opti-
mization [61]. Then, it has been enhanced by J.
Periaux et al. [59, 46, 77] as an excellent way to
dealwithmultidisciplinaryoptimization. Gamethe-
ory seems to be more suitable for multidisciplinary
designthanmostofclassicalmethods. Inparticular,
criterialikethemaximumoftheVonMisesstress,the
dragandthelift,themaximumdisplacement,andthe
fuelconsumption are criteriaof totallydierentna-
ture. Insection3,wesimplyweightedthesecriteria,
but anyweightingoperation is arbitrary. The main
advantageofgametheoryisto workin amulticrite-
riacontext,andeachdisciplineisinchargeofitsown
criterion.
Thenaturalwaytoadaptgametheorytomultidis-
ciplinaryoptimizationistoconsidereachdisciplineas
aplayer.
WeconsidertworealcostfunctionsdenedonA
B:
f
A
: AB ! R
(x;y) 7! f
A (x;y)
(27)
and
f
B
: AB ! R
(x;y) 7! f
B (x;y)
(28)
where AandB aretwopartsofR n
andR m
respec-
tively. PlayerAminimizesthecostfunctionf
A with
respecttox,whileplayerB minimizesthecostfunc-
tionf
B
withrespect toy.
4.1 The Pareto equilibrium
4.1.1 Denition
Thepoint(x
;y
)is Paretooptimal (or aParetoe-
quilibriumpoint)iftheredoesnotexist any(x;y)2
AB suchthat
f
A
(x;y)f
A (x
;y
)
f
B
(x;y)<f
B (x
;y
);
or
f
A
(x;y)<f
A (x
;y
)
f
B
(x;y)f
B (x
;y
):
Thisisequivalenttosaythat thereisnowaytoim-
provef
A
withoutalossofperformanceoff
B
andvice
The set of all Pareto equilibrium points is called
theParetofront.
WhentheParetofrontisaconvexset,onecannd
allitspointswhileminimizingf
=f
A
+(1 )f
B
for all 0 < < 1. For all , there exists a Pareto
equilibriumpoint.
Theteams inchargeofdisciplinesmayusetheset
oftheParetofronttondacompromise.
4.1.2 Example
Letusconsideraverysimpleexample
f
A
=(x 1) 2
+(x y) 2
f
B
=(x 3) 2
+(x y) 2
:
(29)
Onecan nd thePareto equilibrium points bycon-
sideringaconvexcombinationofthetwocriteria
f
=f
A
+(1 )f
B
; (30)
andminimizingf
forall0<<1. Inthiscase,we
have
f
=(x 1) 2
+(1 )(y 3) 2
+(x y) 2
(31)
andthen
@
x f
=2[(x 1)+(x y)]=0
@
y f
=2[(1 )(y 3) (x y)]=0:
(32)
Wenallyobtain
(1+)x y=
x+(2 )y=3(1 ):
(33)
4.2 The Nash equilibrium
4.2.1 Denition
Thepoint(x
;y
)isaNashequilibrium pointif
(
f
A (x
;y
)=inf
x f
A (x;y
)
f
B (x
;y
)=inf
y f
B (x
;y):
(34)
It is straightforward to see that this equilibrium
pointmustsatisytheoptimalitysystem
r
x f
A (x
;y
)=0
r
y f
B (x
;y
)=0
(35)
4.2.2 Example
Weconsideragainexample(29)
f
A
=(x 1) 2
+(x y) 2
f
B
=(y 3) 2
+(x y) 2
;
(36)
andthecorrespondingNashequilibriumisgivenby
@
x f
A
=0
@
y f
B
=0 ()
y=2x 1
x=2y 3 ()
x= 5
3
y= 7
3
(37)
Onemayremarkthatthesolutionof(37)isexactly
thesameasin (33)with =1in therst equation
and=0inthesecondone.
TheNashequilibriumisaninterestingwaytoavoid
arbitrary weighting of cost functions. However, the
compromise,betweendisciplines,ishiddenbyparam-
eters denition. Forexample, thechoice of the pa-
rameters subsets corresponding to discipline 1 and
discipline2isahiddencompromise.
4.3 The Stackelberg equilibrium
4.3.1 Denition
We now assume that one of the two players is the
leaderof thegame, forexampleplayerA. Themul-
tidisciplinaryproblemisformulatedasfollows:
min
x f
A (x;y
x )
wherey
x
=argmin
y f
B (x;y):
(38)
Thisisaneliminationmethodforunknownvariables.
Then,weonlyhaveto minimize
j(x):=f
A (x;y
x
): (39)
Ifweassumethatf
A
isanaerodynamicscostfunc-
tion,forexamplethedrag,andthatf
B
istheweight,
thenyrepresentstheprivateparametersofthestruc-
turalmechanics(numberofspars,position,thickness,
etc.). Each evaluation of f
B
requires the resolution
4.3.2 Example
Westillconsiderthepreviousexample
f
A
=(x 1) 2
+(x y) 2
f
B
=(y 3) 2
+(x y) 2
:
(40)
The Stackelberg equilibrium is exactly the Nash
equilibrium:
y
x
=
(x+3)
2
;
j(x)=(x 1) 2
+
x 3
2
2
:
Theminimizationofj(x)gives
x= 5
3
and y
x
= 7
3
.
4.4 Applicationof the game theory to
the multidisciplinary design
Wesimply haveto consider equations(20) andcost
function (7)
a
1
=A
1 (x;a
2
) (41)
a
2
=A
2 (x;a
1
) (42)
min
x f(x;a
1 (x);a
2
(x)) (43)
g(x;a
1 (x);a
2
(x))0
where theweighted criterionf(x;a
1 (x);a
2
(x))is re-
placed by f
1 (x;a
1 (x);a
2
(x)) for discipline1 and by
f
2 (x;a
1 (x);a
2
(x)) for discipline 2. For example, if
discipline1isstructuralmechanics,f
1
mayrepresent
theweightofthestructure. Ifdiscipline2isaerody-
namics,thenf
2
maybethelift(withaconstrainton
thedrag).
Then, we consider the decomposition introduced
in the frame of theCSSO method (see section 3.5).
But, here the systemlevelcost function is replaced
followingproblem
a
1
=A
1 (x;~a
2
) (44)
~ a
2
=
~
A
2 (x;a
1
) (45)
min
x
1 f
1 (x
1
;x
2
;a
1 (x);a~
2
(x)) (46)
g(x;a
1 (x);a~
2
(x))0
(1 )x
1cur x
1
(1+)x
1cur
; (47)
anddiscipline2solvesthefollowingproblem
~ a
1
=
~
A
1 (x;a
2
) (48)
a
2
=A
2 (x;~a
1
) (49)
min
x
2 f
2 (x
1
;x
2
;~a
1 (x);a
2
(x)) (50)
g(x;~a
1 (x);a
2
(x))0
(1 )x
2cur x
2
(1+)x
2cur
: (51)
One can add the constraints directly in the cost
function in orderto beexactlyin thecontext ofthe
gametheory.
The Pareto front provides a fundamental answer
to the multidisciplinary problems of optimization.
Among all the solutions given by the Pareto front,
one can choose the most relevant solution by tak-
inginto accountimplicitcriteria,such asaesthetics,
manufacturability,...
The Nash equilibrium makes it possible to avoid
themainissueofcompromises.
The Stackelberg equilibrium corresponds exactly
to theoptimization ofinternal (private)variables of
thediscipline.
Gametheoryislittleusedin thecontext ofmulti-
disciplinaryoptimization. However,atvariouslevels,
itcanoerinterestinganswers:
The Pareto equilibrium provides a complete
range of relevant solutions. Among these
solutions, one can choose the best design that
satisessubjectivecriteria.
TheNashequilibriummakesitpossibletoavoid
compromises, even if a hidden compromise lies
inthechoiceofthesubspacesofdiscipline1and
discipline2.
TheStackelbergequilibrium isparticularly well
adaptedto optimizationwith respectto private
variables.
5 Other approaches for collab-
orative design
Insection3wediscussedMDOframeworkinwhicha
complexsystemisdecomposedintosubsystemsbased
ontheaspectoftheinvolvedanalysis (suchasdisci-
pline specic analysis and expertise). These frame-
works are motivated primarily by aerospace indus-
tryinwhichthedesigntask isbasedondisciplinary
analysis,forexamplestructure, uiddynamics,con-
trol etc. In these frameworks allsubsystems are at
equallevelandin principlethereisnorestrictionon
communicationsbetween subsystems. However, de-
signcyclesinindustrieslikeautomotivesectorarenot
disciplinespecicbut areproductobjectdrivenand
maintainmulti-hierarchicalstructurewithrestriction
on communication paths amongst dierent subsys-
tems and levels [52]. The Use of MDO approach-
esdiscussed in section 3would require completere-
structuringoftheorganizationandhence,thissector
follows amulti-level designand optimization frame-
workusingproductdesigntoolslikeanalyticaltarget
cascading(ATC).Wereferto[39,52, 51]forfurther
details on ATC and its applications. Comparative
propertiesofATCandCOandotherapproachesare
discussed in [8]. A new framework based onnested
ATCandCO methodologyformulti-levelsystemsis
describedin[9]
Anotherapproachforcollaborativedesignisbased
on the theory of multi-agent game and collectives
[79,38]. Unlikethehierarchicalstructureconsidered
inearlierframeworksherethesubsytemsareconsid-
eredasplayerswitheach playerhavingstrategiesor
movesforoptimizingitsobjectiveswhile interacting
there moves for optimizing their objectivefunction.
However,whiledoingsoeachplayeralsotriestomin-
imize thesystemobjective function and receivesin-
centivesfor moves that minimize the systemobjec-
tivefunction. Basedonthese incentives,eachplayer
decideshis future moves. An equilibrium isreached
whennoplayercanimproveitsobjectivefunctionfur-
therandthus,alsoresultsinoptimizedsystemobjec-
tive. Thesetechniqueswererstappliedtovarietyof
distributedoptimizationproblemsincludingnetwork
routing,computingresourceallocation,anddatacol-
lection by autonomous rovers [80, 81, 74] however,
onlyrecentlywereappliedto aeronauticalandother
problems[13,14].
6 Conclusion
Most contributions consider collaborative design
fromtheoptimizationpointofview. Wepointedout
thecrucialroleofparametersdenitionasaprelimi-
narystepforMDO.Itisobviousthatthisstephasa
crucialimpactonthenalperformancesofeachdis-
cipline. Wealsothink thatthechoiceofparameters
cannotbedoneinanautomaticway.
Inouropinion, thecollaborativedesignconsistsin
providing an environment of aided design facilities
whereengineersanddesignersstillhavethemostim-
portantrole.
Themainstepsofthecollaborativedesignprocess
are:
Jointdenition ofthepublicparameters,which
aresharedbyalldisciplines.
Creation of disciplinary reduced models allow-
ing real time response for a given value of the
parameters.
Exploration ofthe designspace (set ofpossible
designs)tondthebestdesign.
We also think that there are two main types of
criteriaorcostfunctions:
objectivecriteria,
Thesubjectivecriteria(suchasaesthetics,manufac-
turability,...) arediÆculttohandlebymathemati-
calmodels,andshouldstillbetakenintoaccountby
engineers and designers. Only objective criteriaare
concernedwithmathematicalmodelling.
References
[1] Y. Achdou, Y. A. Kuznetsov, O. Pironneau,
Substructuringpreconditioners fortheQ
1 mor-
tarelementmethod, Numer.Math.,71,pp.419-
449,1995.
[2] N.AlexandrovandS.Kodiyalam,Initial results
ofanMDOmethod evaluation study,Technical
ReportAIAA-98-4884,1998.
[3] N. Alexandrov,ComparativePropertiesof Col-
laborative Optimization and Other Approach-
estoMDO,TechnicalReportICASE-99-209354,
1999.
[4] N. M. Alexandrov and R. M. Lewis, Al-
gorithmic perspectives on problem formula-
tions in MDO, Technical report, 8th A-
IAA/USAF/NASA/ISSMOSymposiumonMul-
tidisciplinary analysis and optimization, AIAA
4719.2000.
[5] N. M. Alexandrov and R. M. Lewis, Analyti-
calandComputationalPropertiesofDistributed
Appraoches to MDO, Technical Report AIAA-
2000-4718,2000.
[6] N. Alexandrov, Analytical and Computational
Aspects of CollaborativeOptimization, Techni-
cal ReportNASA-2000-210104, 2000.
[7] N. Alexandrov, Optimization of Engineering
Systems Governed by Dierential Equations:
Editor'sIntroduction, SIAMActivity Group on
Optimization,11(2),2000.
[8] J. T. Allison, Complex system optimization:
A review of analytical target cascading, col-
laborativeoptimization,andotherformulations,
Master's thesis, Department of Mechanical En-
[9] J. Allison, M. Kokkolaras,M. Zawislak, P. Pa-
palambros, Onthe useof anlaytical target cas-
cading and collaborativeoptimization for com-
plexsystemdesign,6thWorldcongressonStruc-
turalandMultidisciplinaryOptimization,Riode
Janeiro,30 May-03June,Brazil,2005
[10] J. Allison, D. Walsh, M. Kokkolaras, P. Pa-
palambros, M. Cartmell, Analytical target cas-
cadingin aircraftdesign,AIAA paper,2005
[11] A. Aussem, Theorie et application des reseaux
de neurones recurrents et dynamiques a la
prediction,alamodelisationetaucontr^oleadap-
tatifdesprocessusdynamiques,PhDThesis,U-
niversite ReneDescartesParisV,1995.
[12] W. Baur and V. Strassen, The Complexity of
Partial Derivatives, Theoretical Computer Sci-
ence,22,pp.317{330(1983).
[13] S. Bieniawski,D.H. Wolpert, I.Kroo,Discrete,
Continuous, andConstrained Optimization Us-
ingCollectives,AIAAPaper2004-4580, Present-
ed at the 10th AIAA/ISSMO Multidisciplinary
Analysisand Optimization Conference, Albany,
NY,August30-September1,2004.
[14] S.F.Bieniawaski,Distributedoptimizationand
ightcontrolusingcollectives,PhDthesis, Uni-
versityof Stanford,U.S.A., October2005.
[15] R.Braun,P.Gage,I.Kroo,andI.Sobieski,Im-
plementation and performanceissues in collab-
orative optimization, Technical report, AIAA,
1996.
[16] J.Cea,Conceptionoptimaleouidenticationde
forme, calculrapide deladerivee directionnelle
de la fonction co^ut, M. A. A. N., 20(3), pp.
371{402(1986).
[17] M. Cormery, De l'optimisation aerodynamique
vers l'optimisation multidisciplinaire dans un
contexteindustriel,PhDThesis,UniversitePaul
[18] N.Cristianini,J.Shawe-Taylor,Anintroduction
to support vector machines. (and other kernel-
basedlearningmethods),CambridgeUniversity
Press,2000.
[19] M.Daumas,C.Mazenc,andJ.C.Muller,User
Transparent Interval Arithmetic, Technical Re-
port LIP-ENSLyon,1994.
[20] J. C. Gilbert, G. Le Vey, and J. Masse,
La dierentiation automatique de fonctions
representeespardesprogrammes,TechnicalRe-
portINRIA-1557,1991.
[21] J. C. Gilbert, Automatic Dierentiation and
Iterative Processes, Optimization Methods and
software,1,pp.13{21(1992).
[22] A. Griewank, On Automatic Dierentiation,
Mathematical Programming: Recent Develop-
mentsand Applications, M. Iri and K. Tanabe
Eds, Kluwer Academic Publishers, Dordrecht,
1989.
[23] A. Griewank, D. Juedes, J. Srinivasan, and
C. Tyner, ADOL-C, a Package for the Auto-
matic Dierentiation of Algorithms Written in
C/C++,ACMTrans.Math. Software, (1990).
[24] Andreas Griewank, Evaluating Derivatives:
PrinciplesandTechniquesofAlgorithmicDier-
entiation,SIAM,2000
[25] S.GrihonandM.Cormery,MultisciplinaryOp-
timization Appliedto a500Seat Aircraft,Pro-
ceedings 4th ECCOMASConference, K.D. Pa-
pailiouetal.Eds, Athens,Greece,Wiley&Sons
Ltd,1998.
[26] Ph. Guillaume and M. Masmoudi, Derivees
d'ordresuperieuren optimisationde domaines,
C.R.Acad.Sci.Paris,SerieI,315,pp.859{862
(1992).
[27] Ph. Guillaume and M. Masmoudi, Calcul
numeriquedesderiveesd'ordresuperieurencon-
ception optimale de forme, C. R. Acad. Sci.
[28] Ph.GuillaumeandM.Masmoudi,Computation
ofHighOrderDerivativesinOptimalShapeDe-
sign, NumerischeMathematik, 67, pp.231{250
(1994).
[29] Ph. Guillaume and M. Masmoudi, Solution to
the Time-Harmonic Maxwell-s Equations in a
Waveguide, Use of High Order Derivatives for
SolvingtheDiscreteProblem,SIAMJournalon
NumericalAnalysis,(1995).
[30] C.R.Gumbert,J.W.Hou,andP.A.Newman,
Simultaneousaerodynamicandstructuraldesign
optimization (sasdo) for a 2-d wing, Technical
report, AIAA-2001-2527,2001.
[31] Hillstrom, User Guide for Jakef, Technical Re-
port ANL/MCSTM-16, ArgoneNational Labo-
ratory, Argone, II60439,1985.
[32] Control Theory Applications for Optimum De-
signofAerodynamicShapes,Proceedingsofthe
29thIEEEConferenceonDecisionandControl,
Honolulu,Dec.1990,pp.176-179.
[33] D. KalmanandR.Lindell, AutomaticDieren-
tiation in Astrodynamical Modelling, Automat-
ic Dierentiation of Algorithms, A. Griewank,
G.F. CorlissEds, SIAM,1991.
[34] C.T.Kelley,Iterativemethodsforoptimization,
FrontiersinAppliedMathematics, SIAM,1999.
[35] C.T.Kelley,Detectionandremediationofstag-
nationintheNelder-Meadalgorithmusingasuf-
cientdecreasecondition,SIAMJournalofOp-
timization,10(1), pp.43{55(1999).
[36] I.Krooand V. Manning,Collaborativeoptimi-
sation status and directions, Technical Report
AIAA-2000-4721,2000.
[37] I. Kroo, Distributed mulitdisciplinary design
and collaborative optimization,VKI lecture se-
ries on Optimization Methods and Tools for
Multicriteria/Multidisciplinary Design, Novem-
[38] I. Kroo, Collectives and complex design, VK-
I lecture series on Optimization Methods and
Tools for Multicriteria/Multidisciplinary De-
sign,November15-19,2004.
[39] H. M. Kim, Target cascading in optimal sys-
temdesign,PhD thesis, Universityof Michigan,
U.S.A,2001.
[40] K. Kubota, A preprocessor for Fast Automat-
icDierentiation -ApplicationsandDiÆculties
onPracticalProblems,inRIMSKokyuroku648,
Fundamental Numerical Algorithms and their
Software, 1988.
[41] S. Lawrence, A. C. Tsoi, and A. D. Back,
Function approximation with neural networks
and local methods: bias, varianceand smooth-
ness,AustralianConferenceonNeuralNetwork-
s,AustralianNationalUniv.,pp.16{21(1996).
[42] S.Lawrence,C.L.Giles,andA.C.Tsoi,Lesson-
sinneuralnetworktraining: overttingmaybe
harderthanexpected,14th ConferenceonArti-
cialIntelligence,AIAAPress,MenloPark,Cal-
ifornia,pp.540{545(1997).
[43] J. D. Layne, Applying Automatic Dierentia-
tion and Self-Validation Numerical Methods in
SatelliteSimulations,Automatic Dierentiation
of Algorithms, A. Griewank, G. F. Corliss eds,
SIAM,1991.
[44] R. M. Lewis, V. Torczon, and M. W. Tros-
set,Whypatternsearchworks,TechnicalReport
NASA/CR-1998-208966,1998.
[45] Y.Maday,A.Patera,G.Turinici,A prioricon-
vergence theory for reduced-basis approxima-
tionsofsingle-parameterellipticpartialdieren-
tialequations. J.Sci. Comput.17, No.1-4, 437-
446(2002).MSC2000: *65-99.
[46] R. Makinen, P. Neittaanmaki, J. Periaux, and
J. Toivanen, A genetic algorithm for multiob-
jectivedesignoptimizationinaerodynamicsand
'98, Proceedings of the ECCOMAS 98 Confer-
ence, K.D.Papailiou etal.(eds), Wiley, 2,pp.
418{422,Athens,Greece,1998.
[47] J. Markish and K. Willcox, Multidisciplinary
techniques for commercial aircraft system de-
sign,Technical ReportAIAA-2002-5612,2002.
[48] T.Masanao and N. Fujii, SecondOrderNeces-
saryConditionsforDomainOptimizationProb-
lemsinElasticStructures,JournalofOptimiza-
tionTheory andApplications, 72,(1992).
[49] K.I.M.McKinnon,ConvergenceoftheNelder-
Meadalgorithmtoanonstationarypoint,SIAM
Journal on Optimization, 9(1), pp. 148{158
(1998).
[50] B. Mohammadi, O. Pironneau. Applied Shape
Optimization for Fluids. Oxford University
Press2001.
[51] N. Michelena, H. M. Kim, P.Y. Pampalam-
bros, A system partitioning and optimization
approach to target cascading,In Proceedings of
the 12thInternational Conference on Engineer-
ingDesign,Munich, Germany,1999.
[52] N. Michelena, P.Y. Pampalambros, Trends and
challenges in system Design optimization, Pro-
ceedings ofthe International Workshop onMul-
tidisciplinary Design Optimization, Pretoria, S.
Africa,August7-10,2000.
[53] D.W.MonellandW.M.Piland,AerospaceSys-
temsDesigninNASA'sCollaborativeEngineer-
ing, 50th International Astronautical Congress,
Amsterdam, TheNetherlands, 11(2),(1999).
[54] M. Mongeau, H. Karsenty, V. Rouze, and J.-
B.Hiriart-Urruty,Comparisonofpublic-domain
softwareforblackboxglobal optimization, Op-
timizationMethods andSoftware,1999.
[55] J.Morgenstern,Howtocomputefastafunction
andallitsderivatives,avariationonthetheorem
[56] G. M. Ostrovskii, J. M. Volin, and W. W.
Borisov, Uber die Berechnung von Ableitun-
gen, Wissenschaftliche Zeitschrift der Tecnis-
chenHochschulefur Chemie, Leuna Merseburg,
13,pp.382{384(1971).
[57] P.A. LeGresley andJ.J. Alonso.Improvingthe
Performanceof DesignDecomposition Method-
s with POD. 10th AIAA/ISSMO Multidisci-
plinaryAnalysis.AIAA2004-4465,2004.
[58] S.L.Padula,J.J.Korte,H.J.Dunn,andA.O.
Salas, Multidisciplinary Optimization Branch
ExperienceUsingiSightSoftware,TechnicalRe-
portNASA-99-209714,1999.
[59] J. Periaux, Genetic Algorithms and Evolution
Strategyin Engineeringand ComputerScience:
Recent Advances and Industrial Applications,
JohnWiley&Son Ldt,1998.
[60] J.C.Platt,Fast trainingofsupport vectorma-
chines using sequential minimal optimization,
in Advances in Kernel Methods - Support Vec-
tor Learning, (Eds) B. Scholkopf, C. Burges,
and A. J.Smola, MIT Press, Cambridge, Mas-
sachusetts, chapter12,pp185-208,1999.
[61] A. M. Ramos, R. Glowinski, and J. Periaux,
Nashequilibriaforthemultiobjectivecontrolof
linear partial dierential equations, Journal of
OptimizationTheory andApplications, 112(3),
pp.457{498(2002).
[62] J.F.Rodriguez,J.E.Renaud,B.A.Wujek,and
R. V. Trappeta, Trust Region Model Manage-
mentInMultidisciplinaryDesignOptimization,
Journal of Computational and Applied Mathe-
matics,124(1-2),pp.139{154(2000).
[63] N.RostaingandS.Dalmas,AutomaticDieren-
tiationAnalysis and Transformation of Fortran
Program Using a Typed Functional Language,
International Conference on Computing Meth-
odsin AppliedSciencesandEngineering, Paris,
[64] A. O. Salas and J. L. Rogers, A Web-Based
System for Monitoring and Controlling Multi-
disciplinaryDesign,TechnicalReportNASA97-
206287, 1997.
[65] A.O.SalasandJ.C.Townsend,FrameworkRe-
quirement for MDO Application Development,
Technical ReportAIAA-98-4740,1998.
[66] D. Sandy, D. Balkin, and K. Lin, Aneural ap-
proach to response surface methodology, Com-
munications in Statistics, Theory andMethods,
29(9-10),pp.2215-2227(2000).
[67] D.B.Serani,Aframeworkformanagingmod-
elsinnonlinearoptimizationofcomputationally
expensivefunctions,PhDThesis,RiceUniversi-
ty,1998.
[68] J.Shankar,C.Ribbens,R.Haftka,andL.Wat-
son, Computational study of nonhierarchical
decomposition algorithm, Computational Opti-
mizationandApplications, 1993.
[69] I.P.SobieskiandI.M.Kroo,CollaborativeOp-
timization Using Response Surface Estimation,
AIAAJournal,38(10),pp.1931{1938(2000).
[70] B. Speelpenning, Computing Fast Partial
Derivatives of Functions Given by Algorithms,
PhD Thesis, Universityof Illinois,1980.
[71] V.Strassen,AlgebraicComplexityTheory,inJ.
vanLeeuwenEd.,HandbookofTheoreticalCom-
puter Science, Elsevier,Amsterdam,1990.
[72] Z.L.Tang,J.A.DesideriTowardsSelf-Adaptive
Parameterization of Bezier Curves for Air-
foil Aerodynamic Design. INRIA Technical Re-
portRR-4572.October2002.URL:http://www-
sop.inria.fr/rapports/sophia/RR-4572.html
[73] V. Torczon and M. W. Trosset, Using approx-
imations to accelerate engineering design opti-
mization,TechnicalReportNASA-ICASE,1998.
[74] K.Tumer,A.Agogino,D.H.Wolpert,Learning
agents, In Proceedings of the First Internation-
al Joint Conference onAutonomous andMulti-
Agent Systems,Bologna, Italy, July2002.
[75] V.N.Vapnik,TheNatureofStatisticalLearning
Theory,Springer-Verlag,NewYork,1995.
[76] J.L.Walsh, J.C.Townsend, A.O.Salas,J.A.
Samareh,V.Mukhopadhyay,andJ.F.Barthele-
my, Multidisciplinary High-Fidelity Analysis
and Optimization of Aerospace Vehicles, Tech-
nical ReportAIAA-2000-0418,2000.
[77] J.F.Wang,Optimisationdistribueemulticritere
paralgorithmesgenetiques ettheoriedesjeux -
Applicationalasimulationnumeriquedeprob-
lemes d'hypersustentation en aerodynamique,
PhDthesis,UniversiteParisVI, 1992.
[78] R.E.Wengert,ASimpleAutomatic Derivation
EvaluationProgram,Comm. ACM,7,pp.463{
464(1964).
[79] D. H. Wolpert, K. Tumer, An In-
troduction to Collective Intelligence,
Technical Report NASA-ARC-IC-99-63,
NASA Ames Research Center, 1999.
URL:http://ic.arc.nasa.gov/ic/projects/coin
pubs.html.
[80] D. H. Wolpert, K. Tumer, Collective Intel-
ligence, Data Routing, and Braess' Paradox,
JournalofArticialIntelligenceResearch,2002.
[81] D.H. Wolpert,K.Wheeler,K.Tumer, "Collec-
tive Intelligence for Control of Distributed Dy-
namicalSystems,EurophysicsLetters,vol.49is-
sue6,708-714,2000.
[82] J. A. Young, R. D. Anderson, and R. N.
Yurkovich,A descriptionoftheF/A-18E/Fde-
signanddesignprocess,TechnicalReportAIAA-
98-4701,1998.
