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Optimization of Composite Structuresby Estimation of Distribution Algorithms

Paris Research Center, October 5, 2004Laurent [email protected]

Advisors:

Raphael T. Haftka, Department of Mechanical and Aerospace En-gineering, University of Florida

Rodolphe Le Riche, Département Mécanique et Matériaux, ÉcoleNationale Supérieure des Mines de Saint-Étienne

Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 1/50

OutlineRésumé des travaux (en français)



Introduction to composite laminate optimization




Introduction to Estimation of Distribution Algorithms (EDA)




Introduction to Estimation of Distribution Algorithms (EDA)Principles




Introduction to Estimation of Distribution Algorithms (EDA)PrinciplesImportance of the statistical model





Application of a simple EDA to laminate optimization






Introduction of variable dependencies through auxiliary variables andthe Double-Distribution Optimization Algorithm






Introduction of variable dependencies through auxiliary variables andthe Double-Distribution Optimization Algorithm

Conclusions


Résumé des Travaux


Problématique:Optimisation de stratifiés composites

Stratifié composite : empilement de couchesde renforts (fibres) noyées dans la résine(polymère)




Exemples : aéronautique (ex. Boeing 7E7),construction navale, équipements sportifs,. . .





La réponse dépend de l’orientation des fibres






But de l’optimisation = déterminer l’orientation optimale de toutes lescouches pour une application particulière :







maximiser la résistance d’un élément de structure d’un avion







maximiser la résistance d’un élément de structure d’un avionminimiser le poids







maximiser la résistance d’un élément de structure d’un avionminimiser le poidsdéterminer la séquence d’empilement de la coque d’une voiturede course de manière à minimiser les vibrations


État de l’art, objectifs de la thèse

Méthodes traditionnelles basées sur les gradients mais problèmescontinus uniquement et recherche locale




Optimisation stochastique pour globalité (algorithmes génétiques,années 90)





La thèse fait suite à des travaux du groupe de Prof. Haftka:





La thèse fait suite à des travaux du groupe de Prof. Haftka:R. Le Riche : application d’algorithmes évolutionnaires àl’optimisation de structures composites (1994)





La thèse fait suite à des travaux du groupe de Prof. Haftka:R. Le Riche : application d’algorithmes évolutionnaires àl’optimisation de structures composites (1994)B. Liu: optimisation globale de structures composites paralgorithmes multi-niveaux (2001)





La thèse fait suite à des travaux du groupe de Prof. Haftka:R. Le Riche : application d’algorithmes évolutionnaires àl’optimisation de structures composites (1994)B. Liu: optimisation globale de structures composites paralgorithmes multi-niveaux (2001)

Objectifs de la thèse :

Maturation des méthodes évolutionnaires depuis 20 ans (ES, EDA)

Transférer ces nouvelles méthodes à l’optimisation de composites :utilisation plus simpleméthodes plus efficaces


Algorithmes à estimation de distribution

x*

F(x)

x

optimum

x*

F(x)

x

optimum

But : trouver le point de plus élevéd’une “fonction coût” F sur undomaine D



x*

F(x)

x

optimum

x*

F(x)

x

optimum


Difficulté : on ne dispose que d’unbudget limité de N évaluations de lafonction F (on ne peut pas calculertoutes les combinaisons pour choisirla meilleure!)



x*

F(x)

x

optimum

x*

F(x)

x

optimum



On représente notre croyance quel’optimum est dans une région deD par une densité de probabilitép(x)



x*

F(x)

x

optimum

x*

F(x)

x

optimum




p(x) est utilisée pour créer denouveaux points



x*

F(x)

x

optimum

x*

F(x)

x

optimum




p(x) est utilisée pour créer denouveaux pointsles nouveaux points sont utiliséspour mettre à jour p(x)


Enjeux, améliorations proposées et résultats

Éléments critiques de l’algorithme

Adaptation aux problèmes de composites (alphabet non binaire,nombre de variables réduit, évaluations coûteuses)





Représentation de la distribution p(x) : mauvais choix de la forme dep(x) ➫ gaspillage d’évaluations dans des régions médiocres






Solutions proposées







Amélioration du modèle statistique par injection de connaissance surla structure du problème








Mécanismes de contrôle de l’exploration









Résultats









Résultats

Les EDAs sont facilement applicables à l’optimisation de stratifiés









Résultats

Les EDAs sont facilement applicables à l’optimisation de stratifiés

La stratégie proposée conduit à une amélioration de l’efficacité surles problèmes testés


Introduction to Composite LaminateOptimization


Composite Laminates

Composite laminate: structure made of layers(plies) of fibrous material embedded in amatrix


Composite Laminates

Composite laminate: structure made of layers(plies) of fibrous material embedded in a matrix

The fibers (graphite, glass,. . . ) provide themechanical properties, the matrix (polymer)hold the fibers together


Composite Laminates

Composite laminate: structure made of layers(plies) of fibrous material embedded in a matrix

The fibers (graphite, glass,. . . ) provide themechanical properties, the matrix (polymer)hold the fibers together

Applications:aerospace industry (rudder, wing box,flying control surfaces, helicopterblades,. . . )sporting goods (skis, sailing, tennis)wind turbinesmotorsports and performance cars


Optimization of Composite Laminates

Each layer is orthotropic: its mechanical properties depend on thedirection




Overall response F (stiffness, strength, natural frequency, bucklingload,. . . ) of the whole laminate depends on the stacking sequence[θ1, θ2, . . . , θn]:

F = F (θ1, θ2, . . . , θn)





F = F (θ1, θ2, . . . , θn)

Goal of the optimization: maximize F





F = F (θ1, θ2, . . . , θn)


Difficulty:





F = F (θ1, θ2, . . . , θn)


Difficulty:θk discrete (e.g. θk ∈ {0◦, 45◦, 90◦}) ➫ combinatorial problems





F = F (θ1, θ2, . . . , θn)


Difficulty:θk discrete (e.g. θk ∈ {0◦, 45◦, 90◦}) ➫ combinatorial problemsF non-convex in general ➫ many local optima





F = F (θ1, θ2, . . . , θn)


Difficulty:θk discrete (e.g. θk ∈ {0◦, 45◦, 90◦}) ➫ combinatorial problemsF non-convex in general ➫ many local optima

➫ Require specific optimization methods


Laminate Optimization Methods

Gradient-based continuous optimization using ply thicknesses asdesign variables (Schmit and Farshi, 1977) or a penalty approach toforce discreteness of the ply angles (Shin et al., 1990)




➫ can use well-established methods, but no guaranty to find the globaloptimum





Stochastic optimization: Genetic Algorithms (Hajela and Lin, 1992;Le Riche and Haftka, 1993)






➫ stochastic search + discrete variables: can solve the problem directly,but







can be difficult to use (many problem-dependent parameters totune),







can be difficult to use (many problem-dependent parameters totune),and computationally expensive


Purpose of this work

Our goal is to

1. investigate the application of a new class of algorithms called“Estimation of Distribution Algorithms” (EDAs) to the field of laminateoptimization;



Our goal is to


2. propose improvements to EDAs in the context of laminateoptimization:use physics-based knowledge to improve efficiency



Our goal is to


2. propose improvements to EDAs in the context of laminateoptimization:use physics-based knowledge to improve efficiency

3. study the general behavior of EDAs and propose improvements toEDAs that can be used for other fields.


Introduction to Estimation of DistributionAlgorithms


Recall: Genetic Algorithms (GA)

Inspiration: Darwin’s theory of Evolution (“survival of the fittest”)




A species can adapt to an environment becausethose individuals that possess features that give them acompetitive advantage over other individuals are more likelyto have offspring, and therefore to pass on theiradvantageous traits to the next generation.





Application to function optimization:Idea: let a population of candidate solutions evolve to adapt to a taskto perform






Environment → Function F (x) to maximize (“fitness”)






Environment → Function F (x) to maximize (“fitness”)Population of individuals → population of points xi, i = 1, . . . , λ







Natural selection → Selection based on F







Natural selection → Selection based on F

Reproduction → recombination (crossover) and perturbation(mutation) operators


GA Mechanism

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

F(x

)

x

Initial population: uniform


GA Mechanism

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

F(x

)

x


Points obtained byselection ➫ parents


GA Mechanism

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

F(x

)

x



Creation of new points byrecombination(e.g. weighted average inR

n) ➫ children


GA Mechanism

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

F(x

)

x




n) ➫ children

Resulting new population


GA Mechanism

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

20

30

40

50

60

70

F(x

)

x

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

p(x)

Implicit pdf of children




n) ➫ children

Resulting new population

fitness function F

+ selection procedure+ variation operators

➫

implicitprobability distribution p(x)

over the design space


Formalization of GAs ➫ Estimation of Distri-bution Algorithms

Goal: control the way the search distribution p(x) is constructed sothat it learns information about “promising regions”




Principle: express explicitly p(x) and use it to create new pointsin high-fitness areas





Potential advantages:





Potential advantages:Better understanding of the algorithm (based on statisticalprinciples, not nature imitation)➫ improve efficiency





Potential advantages:Better understanding of the algorithm (based on statisticalprinciples, not nature imitation)➫ improve efficiencyPotentially fewer parameters (avoid many ad hoc operators)➫ algorithm easier to use


The Estimation of Distribution Algorithm(EDA)

Initialize P(x)

Create l points

by sampling from P(x)

Select m good points

based on the fitness F

Estimate the distribution

P(x) of the selected points

Initialize P(x)

Create l points








Initialize P(x)

Create l points






Initialize P(x)

Create l points






1. Use created points to inferstatistical information aboutgood regions ➫ p(x)



Initialize P(x)

Create l points






Initialize P(x)

Create l points






1. Use created points to inferstatistical information aboutgood regions ➫ p(x)

2. Use p(x) to create new pointsin promising regions


Estimation of Distribution Algorithm: Illustra-tion

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

F(x

)

x

Initial population: uniformp(x)



0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

F(x

)

x


Points obtained byselection



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

F(x

)

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

p(x)

Gaussian kernels



Estimation of thedistribution of good pointsp(x)(continuous case, kernels,p(x) =const

N

∑Ni=1 exp(− (x−xi)

2

2σ2 ))



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

F(x

)

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

p(x)




N


2

2σ2 ))

Estimated distribution ofgood points p(x)



0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

F(x

)

x




N


2

2σ2 ))

Estimated distribution ofgood points p(x)

New population obtainedby sampling from p(x)


Estimating the distribution of good points

Task: given a sample of µ selected points, infer distribution p(x) of allthe points of comparable fitness (“promising regions”)




Procedure:




Procedure:1. choose a model p̂(x; m1, . . . , mr),




Procedure:1. choose a model p̂(x; m1, . . . , mr),2. find the value of the model parameters mj that maximizes the

likelihood of the good observed points.






The choice of the model is a compromise between






The choice of the model is a compromise between1. the exploitation or accuracy of p, i.e. its ability to learn the

distribution of selected points, and






The choice of the model is a compromise between1. the exploitation or accuracy of p, i.e. its ability to learn the

distribution of selected points, and2. the generalization stability of p in unvisited regions, in particular

its ability to explore them.


Application of a Simple EDA to LaminateOptimization


Simple Model: Independent Variables

General principle in machine learning: in the absence of informationabout the distribution, use simple models (cf. “Occam’s razor”)




Simplest statistical model of selected points: independent variables:

p(x1, x2, . . . , xn) =

n∏

k=1

p(xk)





p(x1, x2, . . . , xn) =

n∏

k=1

p(xk)

Only the marginal distributions of the variables appear in the model:discrete case = frequency of all possible values of each variable





p(x1, x2, . . . , xn) =

n∏

k=1

p(xk)


Proposed by S. Baluja (PBIL, 1994) and H. Mühlenbein (1996) in theUnivariate Marginal Distribution Algorithm (UMDA)





p(x1, x2, . . . , xn) =

n∏

k=1

p(xk)


Proposed by S. Baluja (PBIL, 1994) and H. Mühlenbein (1996) in theUnivariate Marginal Distribution Algorithm (UMDA)

Changes: non-binary alphabet, mutation to compensate forestimation error


Application to Laminates: Frequency problem

Constrained maximization of the first natural frequency of asimply-supported rectangular laminated plate:

maximize f1(θ1, . . . , θ15)

such that νl ≤ νeff ≤ νu

Poisson’s ratio = deformation observed in thetransverse direction when a unit deformation isapplied in the longitudinal direction

L

W




maximize f1(θ1, . . . , θ15)



L

W

θk ∈ {0◦, 15◦, 30◦, 45◦, 60◦, 75◦, 90◦}




maximize f1(θ1, . . . , θ15)



L

W

θk ∈ {0◦, 15◦, 30◦, 45◦, 60◦, 75◦, 90◦}

The constraints are enforced througha penalty approach

020

4060

80100

0

20

40

60

80

100−2000

−1000

0

1000

x1

x2




maximize f1(θ1, . . . , θ15)



L

W

θk ∈ {0◦, 15◦, 30◦, 45◦, 60◦, 75◦, 90◦}

The constraints are enforced througha penalty approach

The constraints create a narrow ridgein the design space

020

4060

80100

0

20

40

60

80

100−2000

−1000

0

1000

x1

x2


Results: reliability of the optimization

Optimum:[904/ ± 75/ ± 602/ ± 455/ ± 305]s



Optimum:[904/ ± 75/ ± 602/ ± 455/ ± 305]s

Compare UMDA to a GAand a hill-climbing algorithm(SHC)



Optimum:[904/ ± 75/ ± 602/ ± 455/ ± 305]s


UMDA and SHC: optimizedparameters, GA: same set-ting as UMDA



Optimum:[904/ ± 75/ ± 602/ ± 455/ ± 305]s



Reliability R: probability of finding the optimum ina given number of function evaluations. 0 2000 4000 6000 8000 10000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

number of evaluations

relia

bilit

y

SHCUMDAGA



Optimum:[904/ ± 75/ ± 602/ ± 455/ ± 305]s



Reliability R: probability of finding the optimum ina given number of function evaluations. 0 2000 4000 6000 8000 10000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


relia

bilit

y

SHCUMDAGA

➫ UMDA outperforms SHC because its distribution approach (global)allows it to handle narrow search spaces

➫ the performance of UMDA is substantially higher than that of GA forthis problem


Improvement of the Statistical Model ofSelected Points through Auxiliary Variables:

the Double-Distribution Optimization Algorithm


Limitations of simple models

Usual assumption: independent variables

p(x1, x2, . . . , xn) = p(x1)p(x2) . . . p(xn)




p(x1, x2, . . . , xn) = p(x1)p(x2) . . . p(xn)

Problem: does not work for problems with strong variable interactions




p(x1, x2, . . . , xn) = p(x1)p(x2) . . . p(xn)


0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

θ1

θ 2

OPTIMUM




p(x1, x2, . . . , xn) = p(x1)p(x2) . . . p(xn)


0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

θ1

θ 2

OPTIMUM

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

θ1

θ 2

OPTIMUM

MAXIMUMPROBABILITY




p(x1, x2, . . . , xn) = p(x1)p(x2) . . . p(xn)


0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

θ1

θ 2

OPTIMUM

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

θ1

θ 2

OPTIMUM

MAXIMUMPROBABILITY

High-probability areas do not coincide with high-fitness regions


Representation of variable dependencies

Complex statistical models have been tried: chain models, treemodels, full Bayesian networks (Pelikan, 1999)




X1 X2 X3




X1 X2 X3

X1

X3

X4 X5

X6




X1 X2 X3

X1

X3

X4 X5

X6

X1 X2

X3 X4

X5




X1 X2 X3

X1

X3

X4 X5

X6

X1 X2

X3 X4

X5

Disadvantages:




X1 X2 X3

X1

X3

X4 X5

X6

X1 X2

X3 X4

X5

Disadvantages:The number of parameters mj to estimate increases rapidly withthe model complexity




X1 X2 X3

X1

X3

X4 X5

X6

X1 X2

X3 X4

X5

Disadvantages:The number of parameters mj to estimate increases rapidly withthe model complexity

➫ the population size needed to estimate these parameters ensure(with a constant confidence) increases with the model complexitybecause flexible models do not generalize well the informationcontained in the sample to other regions


Representation of joint actions of the vari-ables via auxiliary variables

Observation : in many situations, a small number of high ordervariables V = (V1, V2, . . . , Vm) partially determine the objectivefunction




Examples :




Examples :the dimensions of a beam determine its flexural behavior throughthe moment of inertia I,




Examples :the dimensions of a beam determine its flexural behavior throughthe moment of inertia I,the geometry of a vehicle influences its aerodynamic behavior viathe drag coefficient CV ,




Examples :the dimensions of a beam determine its flexural behavior throughthe moment of inertia I,the geometry of a vehicle influences its aerodynamic behavior viathe drag coefficient CV ,the locations of the holes/particles in a porous media affect theflow through the permeability.




Examples :the dimensions of a beam determine its flexural behavior throughthe moment of inertia I,the geometry of a vehicle influences its aerodynamic behavior viathe drag coefficient CV ,the locations of the holes/particles in a porous media affect theflow through the permeability.

These meaningful quantities V reflect joint actions of the variables x

and can be used as auxiliary variables to capture such interactions


Algorithm Principle

The distribution p(x) is represented by a simple model (UMDA)

A simple distribution f(V) is used to introduce variable dependencies

x V

f(V)

V(x1, x2, . . . , xn)

p(x1), p(x2), . . . , p(xn)

Two simple models ⇒ Complex model

f(V)


Algorithm Principle



x V

f(V)

V(x1, x2, . . . , xn)

p(x1), p(x2), . . . , p(xn)


f(V)

Required attributes of theV ’s:


Algorithm Principle



x V

f(V)

V(x1, x2, . . . , xn)

p(x1), p(x2), . . . , p(xn)


f(V)

Required attributes of theV ’s:1. m < n


Algorithm Principle



x V

f(V)

V(x1, x2, . . . , xn)

p(x1), p(x2), . . . , p(xn)


f(V)


2. m does not grow with n


Algorithm Principle



x V

f(V)

V(x1, x2, . . . , xn)

p(x1), p(x2), . . . , p(xn)


f(V)


2. m does not grow with n

3. inexpensive tocompute


The Double-Distribution Optimization Algo-rithm

Initialize P(xi) and f(V)

(Apply mutation)


based on the fitness

Estimate the distributions

P(xi) and f(V) of the selected points

Create l target points

by sampling from f(V)

Create n candidate points

by sampling from P(xi)

Keep l candidate points

closest to the targets points

Initialize P(xi) and f(V)

(Apply mutation)


based on the fitness

Estimate the distributions

P(xi) and f(V) of the selected points









Keep l candidate points

closest to the targets points

Goal: create points whose distribution that reflects both p(x) and f(V):

p(xi) provides a pool of points that have correct marginaldistributions.

f(V) is used as a filter that favors promising regions.

The relative influence of the 2 distributions is adjusted through ν/λ


Application to Laminate Optimization Problems


Case of composite laminates:Auxiliary variables = “Lamination Parameters”

V ≡ Lamination parameters = geometric contribution of the plies tothe stiffness.




E.g. in-plane problem:

A11

A22

A12

A66

= h

U1 U2 U3

U1 −U2 U3

U5 0 −U3

U4 0 −U3

1

V ∗1

V ∗3

h: total laminate thickness, Ui’s: material invariants




E.g. in-plane problem:

A11

A22

A12

A66

= h

U1 U2 U3

U1 −U2 U3

U5 0 −U3

U4 0 −U3

1

V ∗1

V ∗3

h: total laminate thickness, Ui’s: material invariants

Symmetric balanced laminates [±θ1,±θ2, . . . ,±θn]s:

V ∗{1,3} =

2

h

∫ h/2

0

{cos 2θ, cos 4θ}dz =1

n

n∑

k=1

{cos 2θk, cos 4θk}


Representation of the probability distributions

Distribution in the θ-domain




x ≡ θ




x ≡ θ

Discrete univariate model➫ estimate marginal frequen-cies




x ≡ θ


Distribution in the V-domain




x ≡ θ



Continuous variables




x ≡ θ



Continuous variables

Use kernel density estimate:

f(V) =1

µ

µ∑

i=1

K (V − Vi)

In this work, we usedGaussian kernels:

K(u) =1

(2π)d/2σdexp

(

−u

Tu

σ2

)


Estimated Distributions: UMDA and DDOAUMDA: probability concentrated

around (55, 55)Fitness function and selected points

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

θ1

θ 2

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

θ1

θ 2

OPTIMUM



around (55, 55)

DDOA: the mass is distributedalong the “tunnel”

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

θ1

θ 2

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

θ1

θ 2



around (55, 55)

DDOA: the mass is distributedalong the “tunnel”

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

θ1

θ 2

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

θ1

θ 2Sampling from p(θ1), . . . , p(θn) leads to an inaccurate distribution

V-based selection improves the distribution by introducing variabledependencies


Application I: Design of “zero-CTE” laminates

Objective: minimize the longitudinal coefficient of thermal expansion(CTE) subject to a constraint on the first natural vibration frequency,for a 50 in × 15 in plate

minimize |αx|

such that f1 ≥ fmin




minimize |αx|


Constraint enforced through a penalty approach




minimize |αx|



Possible values of the angles: θk ∈ {0◦, 22.5◦, 45◦, 67.5◦, 90◦}




minimize |αx|




Case n = 12: optimum = [904/ ± 67.5/06/ ± 22.56]s




minimize |αx|




Case n = 12: optimum = [904/ ± 67.5/06/ ± 22.56]s

Particularity: response is a function of the lamination parameters only➫ V ’s provide reliable information about the optimum


Comparison with UMDA and GA

0 500 1000 15000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


relia

bilit

y (5

0 ru

ns)

GAUMDADDOA



0 500 1000 15000

0.1

0.2

0.3

0.4

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GAUMDADDOA

GA and UMDA’s progress falls off after 600 analyses



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DDOA reaches high a reliability (90%)



0 500 1000 15000

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DDOA reaches high a reliability (90%)

➫ For this problem, DDOA benefits from the use of auxiliary variables(incorporation of physics-based information improves accuracy ofp(x))


Application II: Strength Maximization

Strength maximization problem:

maximize λs =n

mink=1

(

min

(

ǫult1

ǫ1(k),

ǫult2

ǫ2(k),

γult12

γ12(k)

))

1000 N/m

200 N/m400 N/m

1000 N/m

200 N/m400 N/m




maximize λs =n

mink=1

(

min

(

ǫult1

ǫ1(k),

ǫult2

ǫ2(k),

γult12

γ12(k)

))

1000 N/m

200 N/m400 N/m

1000 N/m

200 N/m400 N/m

Characteristics of this problem:




maximize λs =n

mink=1

(

min

(

ǫult1

ǫ1(k),

ǫult2

ǫ2(k),

γult12

γ12(k)

))

1000 N/m

200 N/m400 N/m

1000 N/m

200 N/m400 N/m

Characteristics of this problem:the V ’s do not capture allthe response:λs = λs(V1, V3, θ1, . . . , θn)




maximize λs =n

mink=1

(

min

(

ǫult1

ǫ1(k),

ǫult2

ǫ2(k),

γult12

γ12(k)

))

1000 N/m

200 N/m400 N/m

1000 N/m

200 N/m400 N/m

Characteristics of this problem:the V ’s do not capture allthe response:λs = λs(V1, V3, θ1, . . . , θn)

many local optima 0

50

1000

50

100

15

20

25

30

35

40

x2x

1



Case n = 12



Case n = 12

θk ∈ {0◦, 22.5◦, 45◦, 67.5◦,90◦



Case n = 12

θk ∈ {0◦, 22.5◦, 45◦, 67.5◦,90◦

Global optimum:[014/ ± 67.55]s, λs = 4.74



Case n = 12

θk ∈ {0◦, 22.5◦, 45◦, 67.5◦,90◦


Parameters: λ = µ = 30, lin-ear ranking selection, pm =0.02



Case n = 12

θk ∈ {0◦, 22.5◦, 45◦, 67.5◦,90◦



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GAUMDADDOA



Case n = 12

θk ∈ {0◦, 22.5◦, 45◦, 67.5◦,90◦



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➫ UMDA and GA locally improve the initial candidate solutions, but failto converge to high-fitness solutions



Case n = 12

θk ∈ {0◦, 22.5◦, 45◦, 67.5◦,90◦



0 500 1000 15000

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GAUMDADDOA

➫ UMDA and GA locally improve the initial candidate solutions, but failto converge to high-fitness solutions

➫ DDOA reliably finds the optimum


Distribution of the solutions found

500 1000 1500 2000 2500 30003

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5


F

UMDA

500 1000 1500 2000 2500 30003

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5


F

GA

500 1000 1500 2000 2500 30003

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5


F

DDOA



500 1000 1500 2000 2500 30003

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5


F

UMDA

500 1000 1500 2000 2500 30003

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5


F

GA

500 1000 1500 2000 2500 30003

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5


F

DDOA

UMDA converges to poor solutions easy to find (e.g.[04/ ± 22.53/ ± 453/ ± 67.5/906]s): the univariate model cannotidentify good regions



500 1000 1500 2000 2500 30003

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5


F

UMDA

500 1000 1500 2000 2500 30003

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5


F

GA

500 1000 1500 2000 2500 30003

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5


F

DDOA


GA correctly converges to the neighborhood of the global optimum,but does not explore it (small variance)



500 1000 1500 2000 2500 30003

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5


F

UMDA

500 1000 1500 2000 2500 30003

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5


F

GA

500 1000 1500 2000 2500 30003

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5


F

DDOA


GA correctly converges to the neighborhood of the global optimum,but does not explore it (small variance)

DDOA focuses the search on high-fitness regions. The two lowerbasins of attraction present in GA and UMDA bearly appear


Diversity preservation mechanisms

Theoretical EDA: infinite population λ




Implementation: finite (small) population ➫ estimation error on p(x)





Observation: tendency to underestimate p in unexplored regions(loss of variable values = “premature convergence”)






➫ Diversity preservation mechanisms must be implemented tocompensate for lost points







Two mechanisms:







Two mechanisms:mutation: a perturbation is applied with probability pm to eachvariable θk of each of the λ created points







Two mechanisms:mutation: a perturbation is applied with probability pm to eachvariable θk of each of the λ created pointslower bound on marginal probabilities: the probability p(θk = cl) isnot allowed to fall below a threshold ǫ


Effect of mutation

UMDA DDOA

0 500 1000 1500 2000 2500 30000

0.05

0.1

0.15

0.2

0.25


Rel

iabi

lity

(95%

,50

runs

)

pm

=0, ε=0p

m=0.005, ε=0

pm

=0.01, ε=0p

m=0.02, ε=0

pm

=0.03, ε=0p

m=0.04, ε=0

pm

=0.05, ε=0

0 500 1000 1500 2000 2500 30000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


Rel

iabi

lity

(95%

,50

runs

)

pm

=0, ε=0p

m=0.005, ε=0

pm

=0.01, ε=0p

m=0.02, ε=0

pm

=0.03, ε=0p

m=0.04, ε=0

pm

=0.05, ε=0

even with a large mutation rate, UMDA does not reliably find theoptimum (probability of obtaining a good point by chance very lowwith n = 12)

mutation greatly improve DDOA’s performance: the auxiliary variablescheme filters out poor candidates created by mutation


Effect of the lower bound on marginal distri-butions

UMDA DDOA

0 500 1000 1500 2000 2500 30000

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Rel

iabi

lity

(95%

,50

runs

)

pm

=0, ε=0p

m=0, ε=0.01

pm

=0, ε=0.02p

m=0, ε=0.04

pm

=0, ε=0.06p

m=0, ε=0.08

0 500 1000 1500 2000 2500 30000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


Rel

iabi

lity

(95%

,50

runs

)

pm

=0, ε=0p

m=0, ε=0.01

pm

=0, ε=0.02p

m=0, ε=0.04

pm

=0, ε=0.06p

m=0, ε=0.08

preventing probabilities to vanish improves UMDA’s reliability but itremains inferior to DDOA.

DDOA greatly benefits from bounding the probabilities. Theperformance is not sensitive to the value of ǫsame explanation as for mutation


Performance with optimized parameters forthe strength problem

Parameter study for GA,UMDA, and DDOA




Let λ, ν, pm, ǫ vary





Best variants:GA: λ = 80, pm = 0.02UMDA: λ = 40, ǫ = 0.06DDOA: λ = 40, ν = 200, ǫ =0.06






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1


relia

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y (9

5%, 5

0 ru

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GAUMDADDOA






0 500 1000 1500 2000 2500 30000

0.1

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1


relia

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GAUMDADDOA

➫ Significant improvement over UMDA (variable dependencies)






0 500 1000 1500 2000 2500 30000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


relia

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5%, 5

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GAUMDADDOA

➫ Significant improvement over UMDA (variable dependencies)

➫ DDOA more efficient than GA even without mutation


General Conclusions


Concluding Remarks

Estimation of Distribution Algorithms are a formalization ofevolutionary algorithms: provide a sound statistical framework


Concluding Remarks


Applicability to laminate optimization was demonstrated:UMDA displayed comparable to higher performance than otherstochastic algorithms (SHC and GA)


Concluding Remarks



A strategy for improving the statistical model of promising regionsthrough auxiliary variables was proposed: the Double-DistributionOptimization Algorithm


Concluding Remarks




Application to laminate optimization showed the efficiency of theapproach for problems with strong variable dependencies+ greater stability to the value of the algorithlm parameters


Concluding Remarks




Application to laminate optimization showed the efficiency of theapproach for problems with strong variable dependencies+ greater stability to the value of the algorithlm parameters

A study of the diversity was conducted: proposed direct control of thedistribution diversity


Future workExtension of DDOA to continuous problems (constraints as auxiliaryvariables): preliminary results are promising



Application to other fields where auxiliary variables are available



Application to other fields where auxiliary variables are available

Detection of failure situations: how to check the validity of p(x) andwhat to do when failure is detected?


Backup Slides


Balanced symmetric laminate

Particular case: bal-anced symmetric laminates[±θ1,±θ2, . . . ,±θn]s

h/2

zn

zn−1

z1

z0

zk

zk−1

n

1

k

θn

−θn

θk

−θk

θ1

−θ1

Mid-plane


Constraint enforcement through a penalty ap-proach

Consider the following optimization problem:

maximize F (x)

such that gj(x) ≥ 0, j = 1, . . . , m


Constraint enforcement through a penalty ap-proach

Consider the following optimization problem:

maximize F (x)

such that gj(x) ≥ 0, j = 1, . . . , m

The penalty approach transforms this constrained problem into anunconstrained problem by decreasing the objective functionproportionally to the constraint violation:

Fp(x) =

{

F (x) if gj(x) ≥ 0, j = 1, . . . , m

F (x) + p minmj=1 (gj(x)) if ∃ k ∈ {1, . . . , m} s.t. gk(x) < 0


Kernel density estimate: principle

f(V ) is continuous, low dimensional (2D or 4D in our problems) ➫Akernel density estimation approach is adopted:

f(V) =1

µ

µ∑

i=1

K (V − Vi)

In this work, we used Gaussian kernels:

K(u) =1

(2π)d/2σdexp

(

−u

Tu

σ2

)

where σ determines the smoothness of the estimate


Kernel density estimate: illustration

2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5

0

0.2

0.4

0.6

0.8

1

V

f(V

)


optimization of composite structures by estimation of ...leriche/defense_anim.pdf · optimization...

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