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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)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 2/50
OutlineRésumé des travaux (en français)
Introduction to composite laminate optimization
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 2/50
OutlineRésumé des travaux (en français)
Introduction to composite laminate optimization
Introduction to Estimation of Distribution Algorithms (EDA)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 2/50
OutlineRésumé des travaux (en français)
Introduction to composite laminate optimization
Introduction to Estimation of Distribution Algorithms (EDA)Principles
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 2/50
OutlineRésumé des travaux (en français)
Introduction to composite laminate optimization
Introduction to Estimation of Distribution Algorithms (EDA)PrinciplesImportance of the statistical model
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 2/50
OutlineRésumé des travaux (en français)
Introduction to composite laminate optimization
Introduction to Estimation of Distribution Algorithms (EDA)PrinciplesImportance of the statistical model
Application of a simple EDA to laminate optimization
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 2/50
OutlineRésumé des travaux (en français)
Introduction to composite laminate optimization
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 2/50
OutlineRésumé des travaux (en français)
Introduction to composite laminate optimization
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
Conclusions
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 2/50
Résumé des Travaux
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 3/50
Problématique:Optimisation de stratifiés composites
Stratifié composite : empilement de couchesde renforts (fibres) noyées dans la résine(polymère)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 4/50
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,. . .
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 4/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 4/50
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 :
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 4/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 4/50
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 avionminimiser le poids
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 4/50
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 avionminimiser le poidsdéterminer la séquence d’empilement de la coque d’une voiturede course de manière à minimiser les vibrations
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 4/50
État de l’art, objectifs de la thèse
Méthodes traditionnelles basées sur les gradients mais problèmescontinus uniquement et recherche locale
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 5/50
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)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 5/50
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:
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 5/50
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:R. Le Riche : application d’algorithmes évolutionnaires àl’optimisation de structures composites (1994)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 5/50
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:R. Le Riche : application d’algorithmes évolutionnaires àl’optimisation de structures composites (1994)B. Liu: optimisation globale de structures composites paralgorithmes multi-niveaux (2001)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 5/50
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: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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 5/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 6/50
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
Difficulté : on ne dispose que d’unbudget limité de N évaluations de lafonction F (on ne peut pas calculertoutes les combinaisons pour choisirla meilleure!)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 6/50
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
Difficulté : on ne dispose que d’unbudget limité de N évaluations de lafonction F (on ne peut pas calculertoutes les combinaisons pour choisirla meilleure!)
On représente notre croyance quel’optimum est dans une région deD par une densité de probabilitép(x)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 6/50
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
Difficulté : on ne dispose que d’unbudget limité de N évaluations de lafonction F (on ne peut pas calculertoutes les combinaisons pour choisirla meilleure!)
On représente notre croyance quel’optimum est dans une région deD par une densité de probabilitép(x)
p(x) est utilisée pour créer denouveaux points
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 6/50
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
Difficulté : on ne dispose que d’unbudget limité de N évaluations de lafonction F (on ne peut pas calculertoutes les combinaisons pour choisirla meilleure!)
On représente notre croyance quel’optimum est dans une région deD par une densité de probabilitép(x)
p(x) est utilisée pour créer denouveaux pointsles nouveaux points sont utiliséspour mettre à jour p(x)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 6/50
Enjeux, améliorations proposées et résultats
Éléments critiques de l’algorithme
Adaptation aux problèmes de composites (alphabet non binaire,nombre de variables réduit, évaluations coûteuses)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 7/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 7/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 7/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 7/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 7/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 7/50
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
Les EDAs sont facilement applicables à l’optimisation de stratifiés
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 7/50
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
Les EDAs sont facilement applicables à l’optimisation de stratifiés
La stratégie proposée conduit à une amélioration de l’efficacité surles problèmes testés
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 7/50
Introduction to Composite LaminateOptimization
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 8/50
Composite Laminates
Composite laminate: structure made of layers(plies) of fibrous material embedded in amatrix
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 9/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 9/50
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 Structures by Estimation of Distribution Algorithms – p. 9/50
Optimization of Composite Laminates
Each layer is orthotropic: its mechanical properties depend on thedirection
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 10/50
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)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 10/50
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)
Goal of the optimization: maximize F
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 10/50
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)
Goal of the optimization: maximize F
Difficulty:
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 10/50
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)
Goal of the optimization: maximize F
Difficulty:θk discrete (e.g. θk ∈ {0◦, 45◦, 90◦}) ➫ combinatorial problems
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 10/50
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)
Goal of the optimization: maximize F
Difficulty:θk discrete (e.g. θk ∈ {0◦, 45◦, 90◦}) ➫ combinatorial problemsF non-convex in general ➫ many local optima
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 10/50
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)
Goal of the optimization: maximize F
Difficulty:θk discrete (e.g. θk ∈ {0◦, 45◦, 90◦}) ➫ combinatorial problemsF non-convex in general ➫ many local optima
➫ Require specific optimization methods
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 10/50
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)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 11/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 11/50
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)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 11/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 11/50
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),
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 11/50
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),and computationally expensive
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 11/50
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;
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 12/50
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;
2. propose improvements to EDAs in the context of laminateoptimization:use physics-based knowledge to improve efficiency
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 12/50
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;
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.
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 12/50
Introduction to Estimation of DistributionAlgorithms
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 13/50
Recall: Genetic Algorithms (GA)
Inspiration: Darwin’s theory of Evolution (“survival of the fittest”)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 14/50
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.
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 14/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 14/50
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”)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 14/50
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”)Population of individuals → population of points xi, i = 1, . . . , λ
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 14/50
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”)Population of individuals → population of points xi, i = 1, . . . , λ
Natural selection → Selection based on F
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 14/50
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”)Population of individuals → population of points xi, i = 1, . . . , λ
Natural selection → Selection based on F
Reproduction → recombination (crossover) and perturbation(mutation) operators
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 14/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 15/50
GA Mechanism
0 0.2 0.4 0.6 0.8 10
10
20
30
40
50
60
70
F(x
)
x
Initial population: uniform
Points obtained byselection ➫ parents
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 15/50
GA Mechanism
0 0.2 0.4 0.6 0.8 10
10
20
30
40
50
60
70
F(x
)
x
Initial population: uniform
Points obtained byselection ➫ parents
Creation of new points byrecombination(e.g. weighted average inR
n) ➫ children
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 15/50
GA Mechanism
0 0.2 0.4 0.6 0.8 10
10
20
30
40
50
60
70
F(x
)
x
Initial population: uniform
Points obtained byselection ➫ parents
Creation of new points byrecombination(e.g. weighted average inR
n) ➫ children
Resulting new population
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 15/50
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
Initial population: uniform
Points obtained byselection ➫ parents
Creation of new points byrecombination(e.g. weighted average inR
n) ➫ children
Resulting new population
fitness function F
+ selection procedure+ variation operators
➫
implicitprobability distribution p(x)
over the design space
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 15/50
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”
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 16/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 16/50
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:
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 16/50
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:Better understanding of the algorithm (based on statisticalprinciples, not nature imitation)➫ improve efficiency
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 16/50
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:Better understanding of the algorithm (based on statisticalprinciples, not nature imitation)➫ improve efficiencyPotentially fewer parameters (avoid many ad hoc operators)➫ algorithm easier to use
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 16/50
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
by sampling from P(x)
Select m good points
based on the fitness F
Estimate the distribution
P(x) of the selected points
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 17/50
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
by sampling from P(x)
Select m good points
based on the fitness F
Estimate the distribution
P(x) of the selected points
1. Use created points to inferstatistical information aboutgood regions ➫ p(x)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 17/50
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
by sampling from P(x)
Select m good points
based on the fitness F
Estimate the distribution
P(x) of the selected points
1. Use created points to inferstatistical information aboutgood regions ➫ p(x)
2. Use p(x) to create new pointsin promising regions
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 17/50
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)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 18/50
Estimation of Distribution Algorithm: Illustra-tion
0 0.2 0.4 0.6 0.8 10
10
20
30
40
50
60
70
F(x
)
x
Initial population: uniformp(x)
Points obtained byselection
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 18/50
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
)
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
Initial population: uniformp(x)
Points obtained byselection
Estimation of thedistribution of good pointsp(x)(continuous case, kernels,p(x) =const
N
∑Ni=1 exp(− (x−xi)
2
2σ2 ))
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 18/50
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
)
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)
Initial population: uniformp(x)
Points obtained byselection
Estimation of thedistribution of good pointsp(x)(continuous case, kernels,p(x) =const
N
∑Ni=1 exp(− (x−xi)
2
2σ2 ))
Estimated distribution ofgood points p(x)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 18/50
Estimation of Distribution Algorithm: Illustra-tion
0 0.2 0.4 0.6 0.8 10
10
20
30
40
50
60
70
F(x
)
x
Initial population: uniformp(x)
Points obtained byselection
Estimation of thedistribution of good pointsp(x)(continuous case, kernels,p(x) =const
N
∑Ni=1 exp(− (x−xi)
2
2σ2 ))
Estimated distribution ofgood points p(x)
New population obtainedby sampling from p(x)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 18/50
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”)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 19/50
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:
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 19/50
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:1. choose a model p̂(x; m1, . . . , mr),
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 19/50
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: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.
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 19/50
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: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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 19/50
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: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 between1. the exploitation or accuracy of p, i.e. its ability to learn the
distribution of selected points, and
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 19/50
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: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 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.
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 19/50
Application of a Simple EDA to LaminateOptimization
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 20/50
Simple Model: Independent Variables
General principle in machine learning: in the absence of informationabout the distribution, use simple models (cf. “Occam’s razor”)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 21/50
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)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 21/50
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)
Only the marginal distributions of the variables appear in the model:discrete case = frequency of all possible values of each variable
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 21/50
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)
Only the marginal distributions of the variables appear in the model:discrete case = frequency of all possible values of each variable
Proposed by S. Baluja (PBIL, 1994) and H. Mühlenbein (1996) in theUnivariate Marginal Distribution Algorithm (UMDA)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 21/50
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)
Only the marginal distributions of the variables appear in the model:discrete case = frequency of all possible values of each variable
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 21/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 22/50
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
θk ∈ {0◦, 15◦, 30◦, 45◦, 60◦, 75◦, 90◦}
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 22/50
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
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 22/50
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
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 22/50
Results: reliability of the optimization
Optimum:[904/ ± 75/ ± 602/ ± 455/ ± 305]s
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 23/50
Results: reliability of the optimization
Optimum:[904/ ± 75/ ± 602/ ± 455/ ± 305]s
Compare UMDA to a GAand a hill-climbing algorithm(SHC)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 23/50
Results: reliability of the optimization
Optimum:[904/ ± 75/ ± 602/ ± 455/ ± 305]s
Compare UMDA to a GAand a hill-climbing algorithm(SHC)
UMDA and SHC: optimizedparameters, GA: same set-ting as UMDA
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 23/50
Results: reliability of the optimization
Optimum:[904/ ± 75/ ± 602/ ± 455/ ± 305]s
Compare UMDA to a GAand a hill-climbing algorithm(SHC)
UMDA and SHC: optimizedparameters, GA: same set-ting as UMDA
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 23/50
Results: reliability of the optimization
Optimum:[904/ ± 75/ ± 602/ ± 455/ ± 305]s
Compare UMDA to a GAand a hill-climbing algorithm(SHC)
UMDA and SHC: optimizedparameters, GA: same set-ting as UMDA
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
➫ 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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 23/50
Improvement of the Statistical Model ofSelected Points through Auxiliary Variables:
the Double-Distribution Optimization Algorithm
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 24/50
Limitations of simple models
Usual assumption: independent variables
p(x1, x2, . . . , xn) = p(x1)p(x2) . . . p(xn)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 25/50
Limitations of simple models
Usual assumption: independent variables
p(x1, x2, . . . , xn) = p(x1)p(x2) . . . p(xn)
Problem: does not work for problems with strong variable interactions
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 25/50
Limitations of simple models
Usual assumption: independent variables
p(x1, x2, . . . , xn) = p(x1)p(x2) . . . p(xn)
Problem: does not work for problems with strong variable interactions
0 10 20 30 40 50 60 70 80 900
10
20
30
40
50
60
70
80
90
θ1
θ 2
OPTIMUM
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 25/50
Limitations of simple models
Usual assumption: independent variables
p(x1, x2, . . . , xn) = p(x1)p(x2) . . . p(xn)
Problem: does not work for problems with strong variable interactions
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 25/50
Limitations of simple models
Usual assumption: independent variables
p(x1, x2, . . . , xn) = p(x1)p(x2) . . . p(xn)
Problem: does not work for problems with strong variable interactions
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 25/50
Representation of variable dependencies
Complex statistical models have been tried: chain models, treemodels, full Bayesian networks (Pelikan, 1999)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 26/50
Representation of variable dependencies
Complex statistical models have been tried: chain models, treemodels, full Bayesian networks (Pelikan, 1999)
X1 X2 X3
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 26/50
Representation of variable dependencies
Complex statistical models have been tried: chain models, treemodels, full Bayesian networks (Pelikan, 1999)
X1 X2 X3
X1
X3
X4 X5
X6
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 26/50
Representation of variable dependencies
Complex statistical models have been tried: chain models, treemodels, full Bayesian networks (Pelikan, 1999)
X1 X2 X3
X1
X3
X4 X5
X6
X1 X2
X3 X4
X5
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 26/50
Representation of variable dependencies
Complex statistical models have been tried: chain models, treemodels, full Bayesian networks (Pelikan, 1999)
X1 X2 X3
X1
X3
X4 X5
X6
X1 X2
X3 X4
X5
Disadvantages:
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 26/50
Representation of variable dependencies
Complex statistical models have been tried: chain models, treemodels, full Bayesian networks (Pelikan, 1999)
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 26/50
Representation of variable dependencies
Complex statistical models have been tried: chain models, treemodels, full Bayesian networks (Pelikan, 1999)
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 26/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 27/50
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 :
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 27/50
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 :the dimensions of a beam determine its flexural behavior throughthe moment of inertia I,
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 27/50
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 :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 ,
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 27/50
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 :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.
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 27/50
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 :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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 27/50
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)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 28/50
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)
Required attributes of theV ’s:
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 28/50
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)
Required attributes of theV ’s:1. m < n
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 28/50
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)
Required attributes of theV ’s:1. m < n
2. m does not grow with n
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 28/50
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)
Required attributes of theV ’s:1. m < n
2. m does not grow with n
3. inexpensive tocompute
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 28/50
The Double-Distribution Optimization Algo-rithm
Initialize P(xi) and f(V)
(Apply mutation)
Select m good points
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)
Select m good points
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)
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
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 29/50
Application to Laminate Optimization Problems
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 30/50
Case of composite laminates:Auxiliary variables = “Lamination Parameters”
V ≡ Lamination parameters = geometric contribution of the plies tothe stiffness.
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 31/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 31/50
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
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}
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 31/50
Representation of the probability distributions
Distribution in the θ-domain
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 32/50
Representation of the probability distributions
Distribution in the θ-domain
x ≡ θ
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 32/50
Representation of the probability distributions
Distribution in the θ-domain
x ≡ θ
Discrete univariate model➫ estimate marginal frequen-cies
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 32/50
Representation of the probability distributions
Distribution in the θ-domain
x ≡ θ
Discrete univariate model➫ estimate marginal frequen-cies
Distribution in the V-domain
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 32/50
Representation of the probability distributions
Distribution in the θ-domain
x ≡ θ
Discrete univariate model➫ estimate marginal frequen-cies
Distribution in the V-domain
Continuous variables
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 32/50
Representation of the probability distributions
Distribution in the θ-domain
x ≡ θ
Discrete univariate model➫ estimate marginal frequen-cies
Distribution in the V-domain
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
)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 32/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 33/50
Estimated Distributions: UMDA and DDOAUMDA: probability concentrated
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 33/50
Estimated Distributions: UMDA and DDOAUMDA: probability concentrated
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 33/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 34/50
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
Constraint enforced through a penalty approach
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 34/50
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
Constraint enforced through a penalty approach
Possible values of the angles: θk ∈ {0◦, 22.5◦, 45◦, 67.5◦, 90◦}
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 34/50
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
Constraint enforced through a penalty approach
Possible values of the angles: θk ∈ {0◦, 22.5◦, 45◦, 67.5◦, 90◦}
Case n = 12: optimum = [904/ ± 67.5/06/ ± 22.56]s
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 34/50
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
Constraint enforced through a penalty approach
Possible values of the angles: θk ∈ {0◦, 22.5◦, 45◦, 67.5◦, 90◦}
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 34/50
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
number of evaluations
relia
bilit
y (5
0 ru
ns)
GAUMDADDOA
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 35/50
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
number of evaluations
relia
bilit
y (5
0 ru
ns)
GAUMDADDOA
GA and UMDA’s progress falls off after 600 analyses
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 35/50
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
number of evaluations
relia
bilit
y (5
0 ru
ns)
GAUMDADDOA
GA and UMDA’s progress falls off after 600 analyses
DDOA reaches high a reliability (90%)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 35/50
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
number of evaluations
relia
bilit
y (5
0 ru
ns)
GAUMDADDOA
GA and UMDA’s progress falls off after 600 analyses
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))
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 35/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 36/50
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
Characteristics of this problem:
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 36/50
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
Characteristics of this problem:the V ’s do not capture allthe response:λs = λs(V1, V3, θ1, . . . , θn)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 36/50
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
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 36/50
Comparison with UMDA and GA
Case n = 12
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 37/50
Comparison with UMDA and GA
Case n = 12
θk ∈ {0◦, 22.5◦, 45◦, 67.5◦,90◦
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 37/50
Comparison with UMDA and GA
Case n = 12
θk ∈ {0◦, 22.5◦, 45◦, 67.5◦,90◦
Global optimum:[014/ ± 67.55]s, λs = 4.74
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 37/50
Comparison with UMDA and GA
Case n = 12
θk ∈ {0◦, 22.5◦, 45◦, 67.5◦,90◦
Global optimum:[014/ ± 67.55]s, λs = 4.74
Parameters: λ = µ = 30, lin-ear ranking selection, pm =0.02
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 37/50
Comparison with UMDA and GA
Case n = 12
θk ∈ {0◦, 22.5◦, 45◦, 67.5◦,90◦
Global optimum:[014/ ± 67.55]s, λs = 4.74
Parameters: λ = µ = 30, lin-ear ranking selection, pm =0.02
0 500 1000 15000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
number of evaluations
relia
bilit
y (9
5%, 5
0 ru
ns)
GAUMDADDOA
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 37/50
Comparison with UMDA and GA
Case n = 12
θk ∈ {0◦, 22.5◦, 45◦, 67.5◦,90◦
Global optimum:[014/ ± 67.55]s, λs = 4.74
Parameters: λ = µ = 30, lin-ear ranking selection, pm =0.02
0 500 1000 15000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
number of evaluations
relia
bilit
y (9
5%, 5
0 ru
ns)
GAUMDADDOA
➫ UMDA and GA locally improve the initial candidate solutions, but failto converge to high-fitness solutions
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 37/50
Comparison with UMDA and GA
Case n = 12
θk ∈ {0◦, 22.5◦, 45◦, 67.5◦,90◦
Global optimum:[014/ ± 67.55]s, λs = 4.74
Parameters: λ = µ = 30, lin-ear ranking selection, pm =0.02
0 500 1000 15000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
number of evaluations
relia
bilit
y (9
5%, 5
0 ru
ns)
GAUMDADDOA
➫ UMDA and GA locally improve the initial candidate solutions, but failto converge to high-fitness solutions
➫ DDOA reliably finds the optimum
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 37/50
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
number of evaluations
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
number of evaluations
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
number of evaluations
F
DDOA
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 38/50
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
number of evaluations
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
number of evaluations
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
number of evaluations
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 38/50
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
number of evaluations
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
number of evaluations
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
number of evaluations
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
GA correctly converges to the neighborhood of the global optimum,but does not explore it (small variance)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 38/50
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
number of evaluations
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
number of evaluations
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
number of evaluations
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
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 38/50
Diversity preservation mechanisms
Theoretical EDA: infinite population λ
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 39/50
Diversity preservation mechanisms
Theoretical EDA: infinite population λ
Implementation: finite (small) population ➫ estimation error on p(x)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 39/50
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”)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 39/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 39/50
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:
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 39/50
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:mutation: a perturbation is applied with probability pm to eachvariable θk of each of the λ created points
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 39/50
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: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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 39/50
Effect of mutation
UMDA DDOA
0 500 1000 1500 2000 2500 30000
0.05
0.1
0.15
0.2
0.25
number of evaluations
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
number of evaluations
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 40/50
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
number of evaluations
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
number of evaluations
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 41/50
Performance with optimized parameters forthe strength problem
Parameter study for GA,UMDA, and DDOA
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 42/50
Performance with optimized parameters forthe strength problem
Parameter study for GA,UMDA, and DDOA
Let λ, ν, pm, ǫ vary
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 42/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 42/50
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
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
number of evaluations
relia
bilit
y (9
5%, 5
0 ru
ns)
GAUMDADDOA
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 42/50
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
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
number of evaluations
relia
bilit
y (9
5%, 5
0 ru
ns)
GAUMDADDOA
➫ Significant improvement over UMDA (variable dependencies)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 42/50
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
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
number of evaluations
relia
bilit
y (9
5%, 5
0 ru
ns)
GAUMDADDOA
➫ Significant improvement over UMDA (variable dependencies)
➫ DDOA more efficient than GA even without mutation
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 42/50
General Conclusions
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 43/50
Concluding Remarks
Estimation of Distribution Algorithms are a formalization ofevolutionary algorithms: provide a sound statistical framework
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 44/50
Concluding Remarks
Estimation of Distribution Algorithms are a formalization ofevolutionary algorithms: provide a sound statistical framework
Applicability to laminate optimization was demonstrated:UMDA displayed comparable to higher performance than otherstochastic algorithms (SHC and GA)
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 44/50
Concluding Remarks
Estimation of Distribution Algorithms are a formalization ofevolutionary algorithms: provide a sound statistical framework
Applicability to laminate optimization was demonstrated:UMDA displayed comparable to higher performance than otherstochastic algorithms (SHC and GA)
A strategy for improving the statistical model of promising regionsthrough auxiliary variables was proposed: the Double-DistributionOptimization Algorithm
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 44/50
Concluding Remarks
Estimation of Distribution Algorithms are a formalization ofevolutionary algorithms: provide a sound statistical framework
Applicability to laminate optimization was demonstrated:UMDA displayed comparable to higher performance than otherstochastic algorithms (SHC and GA)
A strategy for improving the statistical model of promising regionsthrough auxiliary variables was proposed: the Double-DistributionOptimization Algorithm
Application to laminate optimization showed the efficiency of theapproach for problems with strong variable dependencies+ greater stability to the value of the algorithlm parameters
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 44/50
Concluding Remarks
Estimation of Distribution Algorithms are a formalization ofevolutionary algorithms: provide a sound statistical framework
Applicability to laminate optimization was demonstrated:UMDA displayed comparable to higher performance than otherstochastic algorithms (SHC and GA)
A strategy for improving the statistical model of promising regionsthrough auxiliary variables was proposed: the Double-DistributionOptimization Algorithm
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 44/50
Future workExtension of DDOA to continuous problems (constraints as auxiliaryvariables): preliminary results are promising
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 45/50
Future workExtension of DDOA to continuous problems (constraints as auxiliaryvariables): preliminary results are promising
Application to other fields where auxiliary variables are available
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 45/50
Future workExtension of DDOA to continuous problems (constraints as auxiliaryvariables): preliminary results are promising
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?
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 45/50
Backup Slides
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 46/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 47/50
Constraint enforcement through a penalty ap-proach
Consider the following optimization problem:
maximize F (x)
such that gj(x) ≥ 0, j = 1, . . . , m
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 48/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 48/50
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
Optimization of Composite Structures by Estimation of Distribution Algorithms – p. 49/50
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 Distribution Algorithms – p. 50/50
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 Distribution Algorithms – p. 50/50
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 Distribution Algorithms – p. 50/50