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MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
MULTIPLE GRADIENT DESCENTALGORITHM (MGDA)
FOR MULTIOBJECTIVE OPTIMIZATION
Jean-Antoine DésidériINRIA Project-Team OPALE
Sophia Antipolis Méditerranée Centrehttp://www-sop.inria.fr/opale
ISMP 201221st International Synposium on Mathematical Programming,
Berlin, August 19-24, 2012
PDE-Constrained Optimization and Multi-Level/Multi-Grid Methods
ECCOMAS 2012European Congress on Computational Methods in Applied Sciences and
Engineering, Vienna (Austria), September 10-14, 2012
TO16: Computational inverse problems and optimization
1 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Outline1 Foreword2 MGDA
Pareto-stationarityDescent directionMain resultsPracticalitiesMathematical test-cases
3 Aircraft-Wing Aerodynamics4 Automobile Cooling System Design
Problem description and numerical toolsSoftware frameworkNumerical results
5 Domain-Partitioning Model ProblemModel problemQuasi-Newton MethodBasic MGDAMGDA-II
6 MGDA-IIIBird of paradiseAlgorithmProperties
7 MGDA-IV8 Conclusions9 Cooperation and Competition
2 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Foreword: Pareto optimality• Design vector:
Y ∈Υ⊆ RN
• Objective functions to be minimized (reduced):
Ji (Y ) (i = 1, . . . ,n)
Dominance in efficiency(see e.g. K. Miettinen: Nonlinear Multiobjective Optimization, Kluwer Academic Publishers)
Design-point Y 1 dominates design-point Y 2 in efficiency, Y 1 Y 2,iff
Ji (Y 1)≤ Ji (Y 2) (∀i = 1, . . . ,n)
and at least one inequality is strict. (relationship of partial order)
The designer’s Holy Grail: the Pareto setset of "non-dominated design-points" or "Pareto-optimal solutions"
The Pareto frontimage of the Pareto set in the objective function space(bounds the domain of attainable performance)
3 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Pareto front identificationFavorable situation: continuous and convex Pareto frontGradient-based optimization can be used efficiently, if number n ofobjective functions is not too large
Non-convex or discontinuous Pareto fronts do existEvolutionary strategies or GA’s most commonly-used forrobustness: NSGA-II (Deb), PAES
From: A Fast and Elitist Multiobjective Ge-netic Algorithm: NSGA-II, K. Deb, A. Pratap,S. Agarwal, and T. Meyarivan, IEEE Trans-actions in Evolutionary Computation, Vol. 6,No. 2, April 2002.
Nondominated solutions with NSGA-II on KUR (testcase by F. Kursawe, 1990)
n = 3 f1(x) =n−1
∑i=1
−10exp
−√
x2i + x2
i+1
5
f2(x) =n
∑i=1
(|xi |
45 + 5sinx3
i
)4 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Our general objective
Construct a robust gradient-based method for Paretofront identification
Note : in shape optimization, N is often the dimension of a parameterization, meant to be large
as the dimension of a discretization usually is; thus, often, N n. However, the following
theoretical results also apply to cases where n > N which can be the case of other situations in
engineering optimization.
5 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Outline1 Foreword2 MGDA
Pareto-stationarityDescent directionMain resultsPracticalitiesMathematical test-cases
3 Aircraft-Wing Aerodynamics4 Automobile Cooling System Design
Problem description and numerical toolsSoftware frameworkNumerical results
5 Domain-Partitioning Model ProblemModel problemQuasi-Newton MethodBasic MGDAMGDA-II
6 MGDA-IIIBird of paradiseAlgorithmProperties
7 MGDA-IV8 Conclusions9 Cooperation and Competition
6 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Constructing a gradient-basedcooperative algorithm
The basic question :Knowing the gradients, ∇Ji (Y 0), of n criteria, assumed to besmooth functions of the design vector Y ∈ RN , at a given startingdesign-point Y 0, can we define a nonzero vector ω ∈ RN , in thedirection of which the Fréchet derivatives of all criteria have thesame sign, (
∇Ji (Y 0),ω)≥ 0 (∀i = 1,2, ...,n)?
Answer:Yes, if Y 0 is not Pareto-optimal!
Then, −ω is locally a descent direction common to allcriteria.
7 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Notion of Pareto-stationarityClassically, for smooth real-valued functions of Nvariables
Optimality (+ regularity) =⇒ Stationarity
Now, for smooth Rn-valued functions of N variables,which stationarity requirement is to be enforced forPareto-optimality?
Proposition 1: Pareto Stationarity at a design point Y 0
in an open domain B in which the objective-functions are smoothand convex, if there exists a convex combination of the gradientsequal to 0:
∃α = αi(i=1,,n) s.t.n
∑i=1
αi∇Ji (Y 0) = 0 , αi ≥ 0 (∀i) ,n
∑i=1
αi = 1
Then (to be established subsequently):
Pareto-optimality (+ regularity) =⇒ Pareto-stationarity
8 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
General principle
Proposition 2: Minimum-norm element in the convex hullLet ui(i=1,...,n) be a family of n vectors in RN , and U the so-called convexhull of this family, i.e. the following set of vectors :
U =
u ∈ RN / u =
n
∑i=1
αi ui ; αi ≥ 0 (∀i) ;n
∑i=1
αi = 1
Then, U admits a unique element of minimal norm, ω, and the followingholds :
∀u ∈ U : (u,ω)≥ ‖ω‖2
in which (u,v) denotes the usual scalar product of the vectors u and v .
9 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Existence and uniqueness of ω
Existence⇐= U closed. Uniqueness⇐= U convex.To establish uniqueness, let ω1 and ω2 be two realisations of the minimum:
‖ω1‖= ‖ω2‖= Argminu∈U ‖u‖
Then, since U is convex, ∀ε ∈ [0,1], ω1 + εr12 ∈ U (r12 = ω2−ω1) and:
‖ω1 + εr12‖ ≥ ‖ω1‖
Square both sides, and remplace by scalar products:
∀ε ∈ [0,1] : (ω1 + εr12,ω1 + εr12)− (ω1,ω1)≥ 0
Then∀ε ∈ [0,1] : 2ε(r12,ω1) + ε
2(r12, r12)≥ 0
As ε→ 0+, this condition requires that (r12,ω1)≥ 0; but then, for ε = 1:
‖ω2‖2−‖ω1‖2 = 2(r12,ω1) + (r12, r12) > 0
unless r12 = 0, i.e. ω2 = ω1.
Remark: The identification of the element ω is equivalent to the constrained minimization of aquadratic form in Rn , and not RN .
10 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
First consequence
∀u ∈ U : (u,ω)≥ ‖ω‖2
Proof:Let u ∈ U, and r = u−ω.∀ε ∈ [0,1], ω + εr ∈ U (convexity); hence:
‖ω + εr‖2−‖ω‖2 = 2ε(r ,ω) + ε2(r , r)≥ 0
and this requires that:
(r ,ω) = (u−ω,ω)≥ 0
11 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Proof of Pareto-stationarityat Pareto-optimal design-point Y 0 ∈ B (open ball) ⊆ RN
Hyp : Ji (Y ) (∀i ≤ n) smooth and convex in B ; ui = ∇Ji (Y 0) (∀i ≤ n)
Without loss of generality, assume Ji (Y 0) = 0 (∀i). Then:
Y 0 = ArgminY
Jn(Y ) subject to: Ji (Y )≤ 0 (∀i ≤ n−1) (1)
Let Un−1 be the convex hull of the gradients u1,u2, . . . ,un−1 and ωn−1 = Argminu∈Un−1‖u‖.
The vector ωn−1 exists, is unique, and such that:(ui ,ωn−1
)≥ ‖ωn−1‖2 (∀i ≤ n−1).
Two possible situations:1. ωn−1 = 0, and the objective-functions J1,J2, . . . ,Jn−1 satisfy the Pareto stationarity
condition at Y = Y 0. A fortiori, the condition is also satisfied by the whole set ofobjective-functions.
2. Otherwise ωn−1 6= 0. Let ji (ε) = Ji (Y 0− εωn−1) (i = 1, . . . ,n−1) so that ji (0) = 0 andj ′i (0) =−
(ui ,ωn−1
)≤−‖ωn−1‖2 < 0, and for sufficiently small strictly-positive ε:
ji (ε) = Ji (Y 0− εωn−1) < 0 (∀i ≤ n−1)
and Slater’s qualification condition is satisfied for (1). Thus, the LagrangianL = Jn(Y ) + ∑
n−1i=1 λi Ji (Y ) is stationary w.r.t. Y at Y = Y 0:
un +n−1
∑i=1
λi ui = 0
in which λi > 0 (∀i ≤ n−1) since equality constraints hold Ji (Y 0) = 0 (KKT condition).Normalizing this equation results in the Pareto-stationarity condition.
12 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
RN Affine and Vector StructuresNotation
Usually indistinct
When distinction necessary, a dotted symbol is used foran affine subset/subspace
In particular for the convex hull Uthe set of points vectors of a given origin O and equipollent to u ∈ Upoint to.
Rigorously speaking, the term "convex hull" applies to U.
13 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
RN Affine and Vector StructuresConvex hulls
A case where O⊥ /∈ U
U (affine)
U(vector)
A2
O
O⊥u1
u2
u3
u ∈ U
14 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Second consequenceNote that ∀u ∈ U:
u−un =n
∑i=1
αiui −
(n
∑i=1
αi
)un =
n−1
∑i=1
αiun,i (un,i = ui −un)
Thus U⊆ An−1 (or in affine space, U⊆ An−1),where An−1 is a set of vectors pointing to an affine sub-space An−1
of dimension at most n−1.
Define the orthogonal projection O⊥ of O onto An−1
Since−−→OO⊥ ⊥ An−1 ⊇ U:
O⊥ ∈ U⇐⇒ ω =−−→OO⊥
In this case:∀u ∈ U : (u,ω) = const. = ‖ω‖2
15 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Second consequence
ProofSince O⊥ is the orthogonal projection of O onto An−1,
−−→OO⊥ ⊥ An−1, and in particular
−−→OO⊥ ⊥ U.
If additionally O⊥ ∈ U, vector−−→OO⊥ is the minimum-norm element of U: ω =
−−→OO⊥.
In this case, ω can be calculated by ignoring the inequality constraints, ∀i, αi ≥ 0, that areautomatically satisfied by the solution.
Hence, vector ω realizes the minimum of the quadratic form ‖ω‖2 = ‖∑ni=1 αi ui‖2 uniquely
subject to the equality constraint: ∑ni=1 αi = 1.
The Lagrangian is formed with a unique multiplier λ:
L(α,λ) = 12
(n
∑i=1
αi ui ,n
∑i=1
αi ui
)+ λ
(n
∑i=1
αi −1
)
The stationarity condition requires that:
∀i, ∂L∂αi
= (ui ,ω) + λ = 0 =⇒ (ui ,ω) =−λ = const. = ‖ω‖2
By convex combination, the result extends straightforwardly to the whole U.
16 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Application to the case ofgradients
Suppose ui = ∇Ji(Y 0) (∀i); then:
• either ω 6= 0, and all criteria admit positive Fréchet derivativesin the direction of ω (all equal if ω belongs to the interior U)
• or ω = 0, and the current design point Y 0 is Pareto stationary:
∃αii=1,2,...,n (αi ≥ 0,∀i;n
∑i=1
αi = 1) so thatn
∑i=1
αiui = 0
17 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Proposition 3: Common Descent DirectionBy virtue of Propositions 1 and 2 in the particular case where
ui = J ′i =∇Ji(Y 0)
Si
(Si : user-supplied scaling constant; Si > 0), two situations are possible atY = Y 0 :
• either ω = 0, and the design point Y 0 is Pareto-stationary (orPareto-optimal);
• or ω 6= 0, and −ω is a descent direction common to all criteriaJi(x)(i=1,...,n); additionally, if ω⊥ U, the scalar product (u,ω)
(u ∈ U), and the Frechet derivatives (ui ,ω) are all equal to ‖ω‖2.
MGDA : substitute vector ω to the single-criterion gradientin the steepest-descent method.
Proposition 4: ConvergenceCertain normalization provisions being made, in RN , the MultipleGradient Descent Algorithm (MGDA) converges to aPareto-stationary point.
18 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Convergence proof
Define the criteria to be positive (possibly apply an exp-transform), and ∞ at ∞ (possibly add anexploding term outside of the working ball):
∀i, Ji (Y )→ ∞ as ‖Y‖→ ∞.
Assume these criteria to be continuous.
If the iteration stops in a finite number of steps, a Pareto-stationary point has been reached.
Otherwise, the iteration continues indefinitely, generating an infinite sequence of design points,Y k. The corresponding sequences of criteria are infinite, positive and monotone decreasing.They are bounded. Hence, the sequence of iterates, Y k, is itself bounded and it admits asubsequence converging to say Y ?.Necessarily, Y ? is a Pareto-stationary point. To establish this, assume instead that ω?, whichcorresponds to Y ∗, is nonzero. Then for each criterion, there exists a stepsize ρ for which, thevariation is finite. These criteria are in finite number. Hence, the smallest ρ wlll cause a finitevariation to all criteria, and this is in contradiction with the fact that only infinitely-small variationsof the criteria are realized from Y ?.
19 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Remark 1 : practicaldetermination of vector ω
in the convex hull
Problem to be solved in U⊂ RN
ω = Argminu∈U ‖u‖
U =
u ∈ RN / u =
n
∑i=1
αi ui ; αi ≥ 0 (∀i) ;n
∑i=1
αi = 1
Usually, but not necessarily : N ≥ n.
ParameterizationLet
αi = σ2i (i = 1, ...,n)
to satisfy trivially the inequality constraints (αi ≥ 0 ,∀i), and transform the equality constraint,∑
ni=1 αi = 1 into
n
∑i=1
σ2i = 1⇐⇒ σ = (σ1,σ2, ...,σn) ∈ Sn
where Sn is the unit sphere of Rn and precisely not RN .
20 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Determining vector ω (cont’d)The sphere is easily parameterizedusing trigonometric functions of n−1 independent arcs φ1,φ2, ...,φn−1:
σ1 = cosφ1 . cosφ2 . cosφ3 . ... . cosφn−1σ2 = sinφ1 . cosφ2 . cosφ3 . ... . cosφn−1σ3 = 1 . sinφ2 . cosφ3 . ... . cosφn−1
.
.
....
σn−1 = 1 . 1 . ... . sinφn−2 . cosφn−1σn = 1 . 1 . ... . 1 . sinφn−1
(Consider only : φi ∈ [0,π/2] ,∀i and set : φ0 = π
2 .)
Let ci = cos2 φi (i = 1, ...,n)and get:
α1 = c1 . c2 . c3 . ... . cn−1α2 = (1− c1) . c2 . c3 . ... . cn−1α3 = 1 . (1− c2) . c3 . ... . cn−1
.
.
....
αn−1 = 1 . 1 . ... . (1− cn−2) . cn−1αn = 1 . 1 . ... . 1 . (1− cn−1)
(c0 = 0, and ci ∈ [0,1] for all i ≥ 1).
21 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Determining vector ω (end)
The convex hull is thus parameterized in
[0,1]n−1 (n : number of criteria)
independently of N (dimension of design space).
For example, with n = 4 criteria :α1 = c1c2c3
α2 = (1− c1)c2c3
α3 = (1− c2)c3
α4 = (1− c3)
(c1,c2,c3) ∈ [0,1]3
22 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Remark 2 : appropriatenormalization of the gradients
may reveal to be essential : ui = ∇Ji (Y 0)/SiCase n = 2
• Without gradient normalizationThen, ω =
−−→OO⊥ unless the angle < u1,u2 > is acute, and the norms ||u1||, ||u2|| are very
different; in that case, ω is equal the one of smaller norm.
u1u2 ω =−−→OO⊥
u1
u2
ω =−−→OO⊥
u1u2 = ω−−→OO⊥
• With normalization: The equality ω =−−→OO⊥ holds automatically =⇒
EQUAL FRÉCHET DERIVATIVES
General CaseIF the vectors ui are NOT NORMALIZED, those of smallernorms are more influential to determine the direction of vector ω
23 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Normalizations being examinedRecall
ui = J ′i = ∇Ji (Y 0)/Si (Si > 0) =⇒ ω
δY =−ρω =⇒ δJi = (∇Ji ,δY ) =−ρSi (ui ,ω)
(ui ,ω) = const. if ω does not lie on the “edge” of convex hull
• Standard :
ui =∇Ji (Y 0)
‖∇Ji (Y 0)‖• Equal logorithmic variations (whenever ω is not on edge) :
ui =∇Ji (Y 0)
Ji (Y 0)
• Newton-inspired when limJi = 0 :
ui =Ji
‖∇Ji (Y 0)‖2 ∇Ji (Y 0)
• Newton-inspired when limJi 6= 0 :
ui =max
(J(k−1)
i − J(k)i ,δ
)‖∇Ji (Y 0)‖2 ∇Ji (Y 0) , k : iteration no. , δ > 0 (small)
24 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Is standard normalizationsufficient to guarantee that
ω =−−→OO⊥?
(yielding equal Fréchet derivatives)
No, if n > 2 :
A2
u1
u2 u3O⊥
−−→OO⊥ /∈ U
0
Unit Sphere
25 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Experimenting MGDAFrom Adrien ZERBINATI, on-going doctoral thesis
Basic testcase : minimize the functions
f (x ,y) = 4x2 + y2 + xy g(x ,y) = (x−1)2 + 3(y−1)2
26 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Basic testcaseConvergence from different initial design points
DESIGN SPACE FUNCTION SPACE
(0.5,2)
(1.5,2.5)
(-1.5,-2.5)
27 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Fonseca testcase
Minimize 2 fonctions of 3 variables :
f1,2(x1,x2,x3) = 1−exp
(−
3
∑i=1
(xi ±
1√3
)2)
28 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Fonseca testcase (cont’d)Convergence from initial design points over a sphere
DESIGN SPACE FUNCTION SPACE
Continuous but nonconvex front
29 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Fonseca testcase (end)Compare MGDA with Pareto Archived Evolution Strategy
(PAES)
DESIGN SPACE FUNCTION SPACE
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 −1
0
1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
x2
x1
PAES 2x50 & MGDA 6x12
x3
MGDA iterate
PAES generate
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1PAES 2x50 & MGDA 6x12
MGDA iterate
PAES generate
30 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Outline1 Foreword2 MGDA
Pareto-stationarityDescent directionMain resultsPracticalitiesMathematical test-cases
3 Aircraft-Wing Aerodynamics4 Automobile Cooling System Design
Problem description and numerical toolsSoftware frameworkNumerical results
5 Domain-Partitioning Model ProblemModel problemQuasi-Newton MethodBasic MGDAMGDA-II
6 MGDA-IIIBird of paradiseAlgorithmProperties
7 MGDA-IV8 Conclusions9 Cooperation and Competition
31 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Wing-shape optimizationWave-drag minimization in conjunction with
lift maximization
Metamodel-assisted MGDA (Adrien Zerbinati)
• Wing-shape geometryextruded from airfoil geometry (initial airfoil : NACA0012), 10 parameters
• Two-point/two-objective optimization
1 Subsonic Eulerian flow : M∞ = 0.3, α = 8o , for CL maximization2 Transonic Eulerian flow : M∞ = 0.83, α = 2o , for CD
minimization
• Algorithm - Initial database of 40 design points evolves as follows :
1 Evaluate each new point by two 3D Eulerian flow computations2 Construct surrogate models for CL and CD (using the entire
dataset), and perform MGDA to convergence3 Enrich the database with new points, if appropriate
After 6 loops, the database is made of 227 design points(554 flow computations)
32 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Wing-shape optimization2
Convergence of dataset
6 loops
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045−0.85
−0.8
−0.75
−0.7
−0.65
−0.6
−0.55
−0.5
−0.45
drag coefficient
−lif
t coeffic
ient
initial database
1st step
2nd step
3rd step
4th step
5th step
6th step
DRAG (transonic)
-LIFT(subsonic)
33 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Wing-shape optimization3
Low-drag quasi-Pareto-optimal shape
SUBSONIC TRANSONIC
M∞ = 0.3, α = 8o M∞ = 0.83, α = 2o
34 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Wing-shape optimization4
High-lift quasi-Pareto-optimal shape
SUBSONIC TRANSONIC
M∞ = 0.3, α = 8o M∞ = 0.83, α = 2o
35 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Wing-shape optimization5
Intermediate quasi-Pareto-optimal shape
SUBSONIC TRANSONIC
M∞ = 0.3, α = 8o M∞ = 0.83, α = 2o
36 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Outline1 Foreword2 MGDA
Pareto-stationarityDescent directionMain resultsPracticalitiesMathematical test-cases
3 Aircraft-Wing Aerodynamics4 Automobile Cooling System Design
Problem description and numerical toolsSoftware frameworkNumerical results
5 Domain-Partitioning Model ProblemModel problemQuasi-Newton MethodBasic MGDAMGDA-II
6 MGDA-IIIBird of paradiseAlgorithmProperties
7 MGDA-IV8 Conclusions9 Cooperation and Competition
37 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Description of the test case :
In-house CFD Solver(Num3sis) :
• Compressible Navier-Stokes(FV/FE)
• Parallel computing (MPI)
• CPU cost : 4h on 32 cores
• CAD and mesh by GMSH
Test case :• Mach number : 0.1
• Reynolds number : Re ≈ 2800
• Two-criterion shapeoptimization
• Fine mesh (some 700,000points)
38 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
First objective function
Velocity Variance
• Umean =1
Vtot∑
d(cel,Po )≤ε
u(cel)∗Vol(cel)
• σ2vel,x =
1Vtot
∑d(cel,Po )≤ε
(Umean,x −ux (cel))2 ∗Vol(cel)
• σ2 = σ2vel,x + σ2
vel,y + σ2vel,z
39 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Second objective function
Pressure loss• pk = pmean for inlet (k = i), or outlet (k = o)
• uk = ‖u‖mean for in- or out-let
• ∆p = pi −po +ρi u2
i
2− ρou2
o
2
40 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Design parameters
41 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
MGDA on surrogate model
42 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Software framework
43 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Identifying the Pareto set
3
21
Nominal PointInitial database3rd step6th step9th stepInitial Pareto frontFinal Pareto front
Pres
sure
loss
0,5
1
1,5
2
2,5
Velocity variance0,6 0,8 1 1,2 1,4
Figure: MGDA metamodel assisted step by step evolution. Initial and finalnon-dominated sets. 223 solver calls.
44 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Velocity Magnitude Streamlines
1: BEST VELOCITY VARIANCE 2 : NOMINAL SHAPE
SHAPE COMPARISON 3 : BEST PRESSURE LOSS
45 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Velocity magnitude in a sectionjust after the bend :
1: BEST VELOCITY VARIANCE 2 : NOMINAL SHAPE
SECTION POSITION 3 : BEST PRESSURE LOSS
46 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Velocity Magnitude in outputsection :
1 : BEST VELOCITY VARIANCE 2 : NOMINAL SHAPE
SECTION POSITION 3 : BEST PRESSURE LOSS
47 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
NOMINAL versus LOWESTVELOCITY VARIANCE
48 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
NOMINAL versus LOWESTPRESSURE LOSS
49 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
LOWEST VELOCITYVARIANCE versus LOWEST
PRESSURE LOSS
50 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Outline1 Foreword2 MGDA
Pareto-stationarityDescent directionMain resultsPracticalitiesMathematical test-cases
3 Aircraft-Wing Aerodynamics4 Automobile Cooling System Design
Problem description and numerical toolsSoftware frameworkNumerical results
5 Domain-Partitioning Model ProblemModel problemQuasi-Newton MethodBasic MGDAMGDA-II
6 MGDA-IIIBird of paradiseAlgorithmProperties
7 MGDA-IV8 Conclusions9 Cooperation and Competition
51 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Dirichlet problem
Discrete solution by standard 2nd-order finite-differencemethod over 40×40 qradrangular mesh
−∆u = f over Ω = [−1,1]× [−1,1]
u = 0 (Γ = ∂Ω)
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1-0.8
-0.6-0.4
-0.2 0
0.2 0.4
0.6 0.8
1
-0.5
0
0.5
1
1.5
u_h
u_h
’fort.20’ 1.3 1.1 0.9 0.7 0.5 0.3 0.1 0
-0.1 -0.2 -0.3 -0.4 -0.5
x
y
u_h
u_h contour lines 1.3 1.1 0.9 0.7 0.5 0.3 0.1 0
-0.1 -0.2 -0.3 -0.4 -0.5
-1 -0.5 0 0.5 1
x
-1
-0.5
0
0.5
1
y
52 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Domain partitioningFunction value controls at 4 interfaces
yielding 4 sub-domain Dirichlet problems with 2controlled interfaces each
53 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Interface jumps
Formal expression
• over γ1 (0≤ x ≤ 1 ; y = 0):
s1(x) =∂u∂y
(x ,0+)− ∂u∂y
(x ,0−) =
[∂u1
∂y− ∂u4
∂y
](x ,0);
• over γ2 (x = 0 ; 0≤ y ≤ 1):
s2(y) =∂u∂x
(0+,y)− ∂u∂x
(0−,y) =
[∂u1
∂x− ∂u2
∂x
](0,y);
• over γ3 (−1≤ x ≤ 0 ; y = 0):
s3(x) =∂u∂y
(x ,0+)− ∂u∂y
(x ,0−) =
[∂u2
∂y− ∂u3
∂y
](x ,0);
• over γ4 (x = 0 ; −1≤ y ≤ 0):
s4(y) =∂u∂x
(0+,y)− ∂u∂x
(0−,y) =
[∂u4
∂x− ∂u3
∂x
](0,y).
Approximationby 2nd-order one-sided finite-differences
54 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Functionals and matchingcondition
Interface functionals
Ji =∫
γi
12 s2
i w dγi := Ji (v)
that is, explicitly:
J1 =∫ 1
0
12 s1(x)2 w(x)dx J2 =
∫ 1
0
12 s2(y)2 w(y)dy
J3 =∫ 0
−1
12 s3(x)2 w(x)dx J4 =
∫ 0
−1
12 s4(y)2 w(y)dy
(w(t) (t ∈ [0,1]) is an optional weighting function, andw(−t) = w(t).)
Matching condition
J1 = J2 = J3 = J4 = 0
55 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Adjoint problems
Eight Dirichlet sub-problems
∆pi = 0
pi = 0
pi = siw
(Ωi )
(∂Ωi\γi )
(γi )
∆qi = 0
qi = 0
qi = si+1w
(Ωi )
(∂Ωi\γi+1)
(γi+1)
Green’s formula∫γi
si u′i nw =∫
γi
pi n v ′i +∫
γi+1
pi n v ′i+1
and ∫γi+1
si+1 u′i nw =∫
γi
qi n v ′i +∫
γi+1
qi n v ′i+1
56 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Functional gradients
J ′1 =∫ 1
0s1(x)s′1(x)w(x)dx =
∫ 1
0s1(x)
[∂u′1∂y− ∂u′4
∂y
](x ,0)w(x)dx
=∫ 1
0
∂(p1−q4)
∂y(x ,0)v ′1(x)dx +
∫ 1
0
∂p1
∂x(0,y)v ′2(y)dy +
∫ 0
−1
∂q4
∂x(0,y)v ′4(y)dy
J ′2 =∫ 1
0s2(y)s′2(y)w(y)dy =
∫ 1
0s2(y)
[∂u′1∂x− ∂u′2
∂x
](0,y)w(y)dy
=∫ 1
0
∂q1
∂y(x ,0)v ′1(x)dx +
∫ 1
0
∂(q1−p2)
∂x(0,y)v ′2(y)dy +
∫ 0
−1
∂p2
∂y(x ,0)v ′3(x)dx
J ′3 =∫ 0
−1s3(x)s′3(x)w(x)dx =
∫ 0
−1s3(x)
[∂u′2∂y− ∂u′3
∂y
](x ,0)w(x)dx
=−∫ 1
0
∂q2
∂x(0,y)v ′2(y)dy +
∫ 0
−1
∂(q2−p3)
∂y(x ,0)v ′3(x)dx−
∫ 0
−1
∂p3
∂x(0,y)v ′4(y)dy
J ′4 =∫ 0
−1s4(y)s′4(y)w(y)dy =
∫ 0
−1s4(y)
[∂u′4∂x− ∂u′3
∂x
](0,y)w(y)dy
=−∫ 1
0
∂p4
∂y(x ,0)v ′1(x)dx−
∫ 0
−1
∂q3
∂y(x ,0)v ′3(x)dx +
∫ 0
−1
∂(p4−q3)
∂x(0,y)v ′4(y)dy
Conclusion:
J ′i =4
∑j=1
∫γj
Gi,j v ′j dγj (i = 1, ...,4) ; Gi,j : partial gradients.
57 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Other technical details
Dirichlet sub-problemsAll 12 (4 direct, 4×2 adjoint) sub-problems are solved by directinverse (discrete sine-transform)
uh = (ΩX ⊗ΩY ) (ΛX ⊕ΛY )−1 (ΩX ⊗ΩY ) fh
Integralsare approximated by the trapezoidal rule
58 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Reference methodQuasi-Newton Method applied to agglomerated criterion
J =4
∑i=1
Ji
Gradient
∇J =4
∑i=1
∇Ji
Iterationv (`+1) = v (`)−ρ`∇J(`)
Stepsize
δJ = ∇J(`).δv (`) =−ρ`
∥∥∥∇J(`)∥∥∥2
is set to −εJ(`) by fixing
ρ` =εJ(`)∥∥∇J(`)
∥∥2
59 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Quasi-Newton steepest-descentε = 1
Convergence history
Asymptotic Global
1e-06
1e-05
0.0001
0.001
0.01
0.1
0 2 4 6 8 10 12 14 16 18 20
J_1J_2J_3J_4
J = SUM TOTAL
1e-09
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
10
100
0 20 40 60 80 100 120 140 160 180 200
J_1J_2J_3J_4
J = SUM TOTAL
60 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Quasi-Newton steepest descentε = 1
History of gradients over 200 iterations
∂J/∂v1 ∂J/∂v2
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 2 4 6 8 10 12 14 16 18 20
DISCRETIZED FUNCTIONAL GRADIENT OF J = SUM_i J_i W.R.T. V1
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 2 4 6 8 10 12 14 16 18 20
DISCRETIZED FUNCTIONAL GRADIENT OF J = SUM_i J_i W.R.T. V2
61 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Quasi-Newton steepest descentε = 1
History of gradients over 200 iterations
∂J/∂v3 ∂J/∂v4
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 2 4 6 8 10 12 14 16 18 20
DISCRETIZED FUNCTIONAL GRADIENT OF J = SUM_i J_i W.R.T. V3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0 2 4 6 8 10 12 14 16 18 20
DISCRETIZED FUNCTIONAL GRADIENT OF J = SUM_i J_i W.R.T. V4
62 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Quasi-Newton steepest descentε = 1
Discrete solution
-1-0.5
0 0.5
1 -1
-0.5
0
0.5
1
-0.5
0
0.5
1
1.5
u_h
u_h
u_h(x,y) 1.3 1.1 0.9 0.7 0.5 0.3 0.1 0
-0.1 -0.2 -0.3 -0.4 -0.5
x
y
u_h
u_h contour lines 1.3 1.1 0.9 0.7 0.5 0.3 0.1 0
-0.1 -0.2 -0.3 -0.4 -0.5
-1 -0.5 0 0.5 1
x
-1
-0.5
0
0.5
1
y
63 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Basic MGDA
Practical determination of minimum-norm element ω
ω =4
∑i=1
αiui
ui = ∇Ji (Y 0)
using the following parameterization of the convex hull:α1 = c1c2c3
α2 = (1− c1)c2c3
α3 = (1− c2)c3
α4 = (1− c3)
and(c1,c2,c3) ∈ [0,1]3
are discretized by step of 0.01. Best set of coefficients retained.
64 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Basic MGDAStepsize
Iteration
v (`+1) = v (`)−ρ`ω(`)
Stepsize
δJ = ∇J(`).δv (`) =−ρ` ∇J(`).ω
is set to −εJ(`) by fixing
ρ` =εJ(`)
∇J(`).ω
In practice: ε = 1.
65 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Basic MGDAε = 1
Asymptotic Convergence
Unscaled Scaledui = ∇Ji ui = ∇Ji
Ji
66 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Basic MGDAε = 1
Global Convergence
Unscaled Scaledui = ∇Ji ui = ∇Ji
Ji
67 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
First conclusions
Basic MGDA• Scaling essential
• Somewhat deceiving convergence
Who is to blame?• the insufficiently accurate determination of ω;
• the non-optimality of the scaling of gradients;
• the non-optimality of the step-size, the parameter ε beingmaintained equal to 1 throughout;
• the large dimension of the design space, here 76 (4 interfacesassociated with 19 d.o.f.’s).
68 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
MGDA-II : a direct algorithmValid when gradients are linearly independent
User-supplied scaling factors Si(i=1,,n), and scaledgradients
J ′i =∇Ji (Y 0)
Si
(Si > 0; e.g. Si = Ji for logarithmic gradients).
Perform Gram-Schmidt with special normalization
• Set u1 = J ′1
• For i = 2, . . . ,n, set: ui =J ′i −∑k<i ci,k uk
Aiwhere:
∀k < i : ci,k =
(J ′i ,uk
)(uk ,uk
) , and:
Ai =
1−∑k<i
ci,k if nonzero
εi otherwise (εi arbitrary, but small)69 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Consequences
Element ω always in the interior of convex hull
ω =n
∑i=1
αiui αi =1
‖ui‖2∑
nj=1
1
‖uj‖2
=1
1 + ∑j 6=i‖ui‖2
‖uj‖2
< 1
This implies equal projections of ω onto uk
∀k :(uk ,ω
)= αk ‖uk‖2 = ‖ω‖2
and finally: (J ′i ,ω
)= ‖ω‖2 (∀i)
(with a possibly-modified scale S′i = (1 + εi )Si ), that is:the same positive constant.
EQUAL FRÉCHET DERIVATIVESFORMED WITH SCALED GRADIENTS
70 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Geometrical demonstration
u1
u2O
u3
O⊥
u ∈ U
Given J ′i = ∇Ji (Y 0)/Si , perform G.-S. process
to compute ui = [J ′i −∑k<i ci,k uk ]/Ai ; then:
O⊥ ∈ U =⇒ ω =−−→OO⊥ =⇒(
u1,ω)
=(u2,ω
)=(u3,ω
)= ‖ω‖2
J ′i = Aiui + ∑k<i ci,k uk =⇒(J ′i ,ω
)=(Ai + ∑
k<ici,k)
︸ ︷︷ ︸= 1 by
normalization
thru Ai
‖ω‖2 = ‖ω‖2 = const.
71 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
MGDA-II b: automatic rescaleDo not let constant Ai become negative; instead define
Ai = 1−∑k<i
ci,k
only if this number is strictly-positive. Otherwise, use modified scale:
S′i =
(∑k<i
ci,k
)Si
(“automatic rescale”), so that:
c′i,k =
(∑k<i
ci,k
)−1
ci,k , ∑k<i
c′i,k = 1 ,
and set Ai = εi , for some small εi .
Same formal conclusion: the Fréchet derivatives are equal; but the
value is much larger, and (at least) one criterion has been rescaled
72 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Some open questions
• ωII = ω?
• Yes, if n = 2 and angle obtuse
• No, in general.
• Is the following implication
limωII = 0 =⇒ limω = 0
true? (assuming the limit is the result of the MGDA iteration)
In which order should we perform the Gram-Schmidtprocess?
• Etc
73 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
MGDA-IIε = 1
Global convergence
Unscaled Scaledui = ∇Ji ui = ∇Ji
Ji
74 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
MGDA-II bε = 1
Global convergence
Unscaled Scaledui = ∇Ji ui = ∇Ji
Ji
automatic rescale on automatic rescale on
75 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
MGDA-II : Recap
Convergence over 500 iterations
Unscaled Scaledui = ∇Ji ui = ∇Ji
Ji
automatic rescale off automatic rescale on
76 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Outline1 Foreword2 MGDA
Pareto-stationarityDescent directionMain resultsPracticalitiesMathematical test-cases
3 Aircraft-Wing Aerodynamics4 Automobile Cooling System Design
Problem description and numerical toolsSoftware frameworkNumerical results
5 Domain-Partitioning Model ProblemModel problemQuasi-Newton MethodBasic MGDAMGDA-II
6 MGDA-IIIBird of paradiseAlgorithmProperties
7 MGDA-IV8 Conclusions9 Cooperation and Competition
77 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
MGDA-IIIBird of paradise (Strelitzia reginae)
78 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
MGDA-IIIBird of paradise (Strelitzia reginae)
79 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
MGDA-IIIBird of paradise (Strelitzia reginae)
u1
u2
ω
80 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
MGDA-IIIBird of paradise (Strelitzia reginae)
ω
81 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
MGDA-III - definition1
Start from scaled gradients, J ′i , and duplicates, gi,J ′i =
∇Ji (Y 0)
Si(Si : user-supplied scale)
and proceed in 3 steps: A,B and C.
A: Initialization
• Set
u1 := g1 = J ′k / k = Argmaxi minj
(J ′j ,J
′i
)(J ′i ,J
′i
)(justified afterwards)
• Set n×n lower-triangular array c = ci,j (i ≥ j) to 0.
• Set, conservatively, I := n(expected number of computed orthogonal basis vectors).
• Assign some appropriate value to a cut-off constant a :0≤ a < 1.
(Note: main diagonal of array c is to contain cumulative row-sums.)82 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
MGDA-III - definition2
B: Main G.-S. loop; for i = 2,3, . . . , (at most) n, do:
1. Calculate the i−1st column of coefficients:
cj,i−1 =
(gj ,ui−1
)(ui−1,ui−1
) (∀j = i, . . . ,n)
and update the cumulative row-sums:
cj,j := cj,j + cj,j−1 = ∑k<i
cj,k (∀j = i, . . . ,n)
2. Test:• If the following condition is satisfied
cj,j > a (∀j = i, . . . ,n)
set I := i−1, and interrupt the Gram-Schmidt process(go to 3.).
• Otherwise, compute next orthogonal vector ui as follows:
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MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
MGDA-III - definition3
- Identify index ` = Argminj cj,j / i ≤ j ≤ n(Note that c`,` ≤ a < 1.)
- Permute information associated with i and ` :g-vectors gi g`, rows i and ` of array c and cumulative row-sums ci,i c`,`.
- Set Ai = 1− ci,i ≥ 1−a > 0 (ci,i = former-c`,` ≤ a), and calculate
ui =gi −∑k<i ci,k uk
Ai
(in which gi = former-g`, ci,k = former-c`,k ).
- If ui 6= 0, return to 1. with incremented i; otherwise:
gi = ∑k<i
ci,k uk = ∑k<i
c′i,k gk
where the c′i,k are calculated by backward substitution.Then, if c′i,k ≤ 0 (∀k < i):
Pareto-stationarity detected: STOP MGDA iteration;otherwise (ambiguous exceptional case):
STOP G.-S. process; compute original ω and go to C.
84 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
MGDA-III - definition4
3. Calculate ω as the minimum-norm element in the convex hullof u1,u2, . . . ,uI:
ω =I
∑i=1
αiui 6= 0
αi =1
‖ui‖2∑
Ij=1
1
‖uj‖2
=1
1 + ∑j 6=i‖ui‖2
‖uj‖2
(Note that here all computed ui 6= 0, and 0 < αi < 1.)
C: If ‖ω‖< TOL, STOP MGDA iteration; otherwise,perform descent step and return to B.
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MGDA applied tocompressible aerodynamics
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Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
MGDA-III - consequencesIf I = n :MGDA-III = MGDA-II b + automatic ordering in Gram-Schmidtprocess making ’rescale’ unnecessary since, by construction
∀i : Ai ≥ 1−a > 0
Otherwise, I < n (incomplete Gram-Schmidt process):
• First I Fréchet derivatives:(gi ,ω
)=(ui ,ω
)= ‖ω‖2 > 0 (∀i = 1, . . . , I)
• Subsequent ones (i > I):
ω =I
∑j=1
αjuj gi =I
∑k=1
ci,k uk + vi vi ⊥ u1,u2, . . . ,uI
⇓(gi ,ω
)=
I
∑k=1
ci,k(uk ,ω
)=
I
∑k=1
ci,k ‖ω‖2 = ci,i ‖ω‖2 > a‖ω‖2 > 086 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Choice of g1Justification
At initialization, we have set
u1 := g1 = J ′k / k = Argmaxi minj
(J ′j ,J
′i
)(J ′i ,J
′i
)This was equivalent to maximizing c2,1 = c2,2, that is,maximizing the least cumulative row-sum, at firstestimation.
(Recall that the favorable situation is when all cumulative row-sums are positive, or > a.)
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MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Expected benefits
Automatic ordering and rescale
When gradients exhibit a general trend, ω is found infewer steps, and has a larger norm
Larger ‖ω‖ implies larger directional derivatives, andgreater efficiency of descent step(
gi ,ω
‖ω‖)
= ‖ω‖ or a‖ω‖
88 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Outline1 Foreword2 MGDA
Pareto-stationarityDescent directionMain resultsPracticalitiesMathematical test-cases
3 Aircraft-Wing Aerodynamics4 Automobile Cooling System Design
Problem description and numerical toolsSoftware frameworkNumerical results
5 Domain-Partitioning Model ProblemModel problemQuasi-Newton MethodBasic MGDAMGDA-II
6 MGDA-IIIBird of paradiseAlgorithmProperties
7 MGDA-IV8 Conclusions9 Cooperation and Competition
89 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Adressing the question ofscaling
When Hessians are known• For objective function Ji (Y ) alone, Newton’s iteration writes
Y 1 = Y 0−pi Hipi = ∇Ji (Y 0)
• Split pi into orthogonal components
pi = qi + ri
where: qi =
(pi ,∇Ji (Y 0)
)‖∇Ji (Y 0)‖2 ∇Ji (Y 0) and ri ⊥ ∇Ji (Y 0)
• Define scaled gradientJ ′i = qi
• Apply MGDA-III
Equivalently, set scaling factor as follows: Si =‖∇Ji (Y 0)‖2(pi ,∇Ji (Y 0)
)90 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
VariantMGDA-IV b
Use BFGS iterative estimates in place of exact Hessians
k : MGDA iteration index
(∀i = 1, . . . ,n) H(0)i = Id
H(k+1)i = H(k)
i −1
s(k)T H(k)i s(k)
H(k)i s(k)s(k)T
H(k)T
i +1
z(k)T
i s(k)z(k)
i z(k)T
i
in which:
s(k) = Y (k+1)−Y (k)
z(k)i = ∇Ji (Y (k+1))−∇Ji (Y (k))
91 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Recommended Step-Size
Ji (Y 0−ρω) = Ji (Y 0)−ρ(∇Ji (Y 0),ω
)+ 1
2 ρ2(Hiω,ω
)+ . . . .
|δJi | := Ji (Y 0)− Ji (Y 0−ρω) := Aiρ− 12 Biρ
2 ,
where: Ai = Siai , Bi = Sibi , and:
ai =(J ′i ,ω
)bi =
(Hiω,ω
)/Si .
If the scales Si are truly relevant for the objective-functions
ρ? = Argmax
ρmin
iδi .
where:
δi =|δJi |Si
= aiρ− 12 biρ
2 .
92 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Recommended Step-Size2
I = n
ρ? = ρ
?I :=
‖ω‖2
bI,
where bI = maxi≤I bi is assumed to be positive.
I < n
ρ?II :=
a‖ω‖2
bII,
where bII = maxi>I bi is assumed to be positive.Also
ρ× =2(1−a)‖ω‖2
bI−bII
at which point the two bounding parabolas intersect:
‖ω‖2ρ− 1
2 bIρ2 = a‖ω‖2
ρ− 12 bIIρ
2 .
93 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Recommended Step-Size3
a) I = n
||ω||2δ
ρρ?I
b) I < n, ρ× > max(ρ?I ,ρ
?II )
||ω||2
a||ω||2
δ
ρρ?I ρ?
II ρ×
c) I < n, ρ× ∈ (ρ?I ,ρ
?II )
||ω||2
a||ω||2
δ
ρρ?I ρ× ρ?
II
d) I < n, ρ× < min(ρ?I ,ρ
?II )
||ω||2
δ
ρρ?I
ρ×
ρ?II
94 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Recommended Step-Sizeend
If I = n, ρ? = ρ?I .
If I < n, ρ? is the element of the triplet ρ?I ,ρ
?II ,ρ×
which separates the other two.
95 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Outline1 Foreword2 MGDA
Pareto-stationarityDescent directionMain resultsPracticalitiesMathematical test-cases
3 Aircraft-Wing Aerodynamics4 Automobile Cooling System Design
Problem description and numerical toolsSoftware frameworkNumerical results
5 Domain-Partitioning Model ProblemModel problemQuasi-Newton MethodBasic MGDAMGDA-II
6 MGDA-IIIBird of paradiseAlgorithmProperties
7 MGDA-IV8 Conclusions9 Cooperation and Competition
96 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
CONCLUSIONSTheory
Multiple-Gradient Descent Algorithm
• Notion of Pareto stationarity introduced to characterize"standard Pareto-optimal points"
• Basic MGDA : permits to identify a descent direction commonto arbitrary number of criteria, knowing the local gradients innumber at most equal to the number of design parameters
• Proof of convergence of MGDA to Pareto-stationary points
• Capability to identify the Pareto front demonstrated formathematical test-cases; non-convexity of Pareto front not aproblem
• MGDA-II : a direct procedure, faster and more accurate, whengradients are linearly independent; variant MGDA-II b (withautomatic rescale) recommended
• On-going research: scaling, preconditioning, step-sizeadjustment, what to do to extend MGDA-II to cases oflinearly-dependent gradients, . . .
97 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
CONCLUSIONSApplications
Implementation and first experiments
• Validation with analytical test cases and comparison with aPareto capturing method (PAES)
• Extension to Meta-Model-Assisted MGDA
Aircraft-Wing Aerodynamics and AutomobileCooling-System Design
• Successful design experiments with 3D compressible flow(Euler and Navier-Stokes)
• On-going development of meta-model-assisted MGDA andtesting of 3D external aerodynamics case to be presented atECCOMAS 2012, Vienna, Austria)
98 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
CONCLUSIONSApplications
Domain-partitioning model problemThe method works, but it is not as cost-efficient as the standardquasi-Newton method applied to the agglomerated criterion; but thetest-case was peculiar in several ways:
• Pareto front and set reduced to a singleton
• large dimension of the design space (4 criteria, 76 design variables)
• robust procedure to adjust the step-size needed
• determination of ω in the basic method should be made more accurately, perhaps usingiterative refinement
• scaling of gradients found important but never really analyzed completely (logarithmicgradients appropriate for vanishing criteria)
• general preconditioning not yet clear (on-going)
The MGDA-II direct variant found faster and more robust; withautomatic rescaling procedure on, MGDA-II b seems to exhibitquadratic convergence.
99 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Outline1 Foreword2 MGDA
Pareto-stationarityDescent directionMain resultsPracticalitiesMathematical test-cases
3 Aircraft-Wing Aerodynamics4 Automobile Cooling System Design
Problem description and numerical toolsSoftware frameworkNumerical results
5 Domain-Partitioning Model ProblemModel problemQuasi-Newton MethodBasic MGDAMGDA-II
6 MGDA-IIIBird of paradiseAlgorithmProperties
7 MGDA-IV8 Conclusions9 Cooperation and Competition
100 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Flirting with the Pareto frontMGDA is combined with a locally-adapted Nash game
based on a territory splitting that preserves tosecond-order the Pareto-stationarity condition
Cooperation AND Competition
• COOPERATION (MGDA) :The design point is steered to the Pareto front.
• COMPETITION (Nash game) : at start (from the Pareto front) :α1∇J1 +α2∇J2 = 0; then let : JA = α1J1 +α2J2, et JB = J2, andadapt territory-splitting to best preserve JA; then :the trajectory remains tangent to the Pareto front.
101 / 102
MGDA applied tocompressible aerodynamics
J.-A. Désidéri
Foreword
MGDA
Pareto-stationarity
Descent direction
Main results
Practicalities
Mathematical test-cases
Aircraft-Wing Aerodynamics
Automobile Cooling SystemDesign
Problem description and numericaltools
Software framework
Numerical results
Domain-Partitioning ModelProblem
Model problem
Quasi-Newton Method
Basic MGDA
MGDA-II
MGDA-III
Bird of paradise
Algorithm
Properties
MGDA-IV
Conclusions
Cooperation and Competition
Some references
Reports from INRIA Open Archive:
• M. G. D. A., INRIA Research Report No. 6953 (2009),Revised version, October 2012(Open Archive : http://hal.inria.fr/inria-00389811)
• Related INRIA Research Reports:Nos. 7667 (2011), 7922 (2012), 7968 (2012), 8068 (2012)
Other• C. R. Acad. Sci. Paris, Ser. I 350 (2012) 313-318
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