[PPT] - T H S E OPTIMISATION GLOBALE SEMI-DTERMINISTE ET APPLICATIONS PowerPoint Presentation

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09 Juin 2006 Soutenance de thèse - p. 1/19

T H È S E OPTIMISATION GLOBALE SEMI-DÉTERMINISTE ET APPLICATIONS INDUSTRIELLES

Soutenue par: Ivorra Benjamin Sous la direction de: Bijan Mohammadi À: l’Université Montpellier II

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Outlines

PART I: Global Optimization Methods PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 2/19

Outlines

Global Optimization Methods
BVP formulation of optimization problems
Implementation: 1st and 2nd order methods
Genetic algorithms and dynamical systems

SLIDE 3

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 2/19

Outlines

Global Optimization Methods
BVP formulation of optimization problems
Implementation: 1st and 2nd order methods
Genetic algorithms and dynamical systems
Industrial Applications
Shape Optimization of a Fast-Microfluidic Mixer Device
Multichannel Optical Filters Design
Portfolio Optimization Under Constraints

SLIDE 4

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 2/19

Outlines

Global Optimization Methods
BVP formulation of optimization problems
Implementation: 1st and 2nd order methods
Genetic algorithms and dynamical systems
Industrial Applications
Shape Optimization of a Fast-Microfluidic Mixer Device

◆ Modeling ◆ Numerical results ◆ Comparison with experimental results

Multichannel Optical Filters Design
Portfolio Optimization Under Constraints

SLIDE 5

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 2/19

Outlines

Global Optimization Methods
BVP formulation of optimization problems
Implementation: 1st and 2nd order methods
Genetic algorithms and dynamical systems
Industrial Applications
Shape Optimization of a Fast-Microfluidic Mixer Device
Multichannel Optical Filters Design
Portfolio Optimization Under Constraints
Conclusions and perspectives

SLIDE 6

Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 3/19

PART I: Global Optimization Methods

SLIDE 7

Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 4/19

Problem

min

x∈Ωad

J(x)

Where:

x is the optimization parameter
Ωad ∈ I

RN is the admissible space

SLIDE 8

Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 4/19

Problem

min

x∈Ωad

J(x)

Where:

x is the optimization parameter
Ωad ∈ I

RN is the admissible space Assumptions:

J ∈ C2(Ωad, I

R)

J coercive
Jm denotes: the minimum of J or a low value

SLIDE 9

Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 5/19

BVP formulation

Many minimization algorithms can be seen as discretizations

f dynamical systems with initial conditions.

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 5/19

BVP formulation

Many minimization algorithms can be seen as discretizations

f dynamical systems with initial conditions.

Solve numerically the optimization problem with one of those algorithms (core optimization method) ⇔ Solve this BVP:

First or second order initial value problem

mint∈[0,Z](|J(x(t)) − Jm|) < ǫ where Z ∈ I R is a time and ǫ is the approximation precision.

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 5/19

BVP formulation

Many minimization algorithms can be seen as discretizations

f dynamical systems with initial conditions.

Solve numerically the optimization problem with one of those algorithms (core optimization method) ⇔ Solve this BVP:

First or second order initial value problem

mint∈[0,Z](|J(x(t)) − Jm|) < ǫ where Z ∈ I R is a time and ǫ is the approximation precision. This BVP is over-determined: more conditions than derivatives.

SLIDE 12

Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 6/19

General method for BVP resolution

Idea: Remove the over-determination: One initial condition is considered as a variable v.

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 6/19

General method for BVP resolution

Idea: Remove the over-determination: One initial condition is considered as a variable v. Objective: Find a v solving BVP .

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 6/19

General method for BVP resolution

Idea: Remove the over-determination: One initial condition is considered as a variable v. Objective: Find a v solving BVP . How ? We consider a function h : Ωad → I R: h(v) = min

t∈[0,Z](J(x(t, v)) − Jm)

Solve:

min

v∈Ωad

h(v)

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 7/19

Implementation: 1st order dynamical system

BVP is rewritten as:      M(ζ, x(ζ))xζ = −d(x(ζ)) x(0) = x0 mint∈[0,Z](|J(x(t)) − Jm|) < ǫ Where:

x0 ∈ Ωad the initial condition
ζ is a fictitious time
d a direction in Ωad
M is an operator

Idea: find a x0 solving BVP (The problem is admissible).

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 7/19

Implementation: 1st order dynamical system

Geometrical interpretation of h, using steepest descent:

J h

X h(X)

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 7/19

Implementation: 1st order dynamical system

Single layer algorithm A1(X1):

J h

X1 h(X1)

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 7/19

Implementation: 1st order dynamical system

Single layer algorithm A1(X1):

J h

X1 h(X1) X2 h(X2)

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 7/19

Implementation: 1st order dynamical system

Single layer algorithm A1(X1): X3 = X2 − J(X2)

X2−X1 J(X2)−J(X1)

J h

X1 h(X1) X2 h(X2) X3 h(X3)

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 7/19

Implementation: 1st order dynamical system

Single layer algorithm A1(X1): Problems:

One dimensional search.
The line search minimization might fail.

Idea: Add an external level to the algorithm A1

Find minv′∈Ωad h′(v′) = h(A1(v′))

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 7/19

Implementation: 1st order dynamical system

2D benchmark function:

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 7/19

Implementation: 1st order dynamical system

Two-layers algorithm A2(X1):

X1 X3 X2=A1(X1) h(X1) h(X2) h(X3)

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 7/19

Implementation: 1st order dynamical system

Two-layers algorithm A2(X1):

X1 X3 X2=A1(X1) h(X1) h(X2) h(X3) X’1=A1(X’1) X’2 X’3 h(X’1) h(X’2) h(X’3)

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 7/19

Implementation: 1st order dynamical system

Two-layers algorithm A2(X1):

X1 X3 X2=A1(X1) h(X1) h(X2) h(X3) X’1=A1(X’1) X’2 X’3 h(X’1) h(X’2) h(X’3) X’’1 X’’3=A1(x’’1) X’’2 h(X’’1)=h(X’’3) h(X’’2)

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 7/19

Implementation: 1st order dynamical system

We can build recursively a i-layers algorithm Ai by considering:

min

vi∈Ωad

hi(vi)

with:

hi(vi) = hi−1(Ai−1(vi))
h1 = h′
h0 = h

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 7/19

Implementation: 1st order dynamical system

Example:

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 8/19

Implementation: 2nd order dynamical system

BVP is rewritten as:      ηxζζ(ζ) + M(ζ, x(ζ))xζ(ζ) = −d(x(ζ)), x(0) = x0, xζ(0) = xζ,0 mint∈[0,Z](|J(x(t)) − Jm|) < ǫ where η << 1. Idea:

find a x0 solving BVP (admissible)
r
find a xζ,0 solving BVP (admissible ?)

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 8/19

Implementation: 2nd order dynamical system

Theorem: (With Patrick Redont & Jean Paul Dufour (UM II)) Let J : Rn → R , C2 such that: minRn J exist and is reached at xm ∈ Rn. Then ∀(x0, δ) ∈ Rn × R+

∗ , ∃(σ, γ) ∈ Rn × R such that the

solution of the following dynamical system:      ηxζζ(ζ) + xζ(ζ) = −∇J(x(ζ)) x(0) = x0 xζ(0) = σ Pass at time γ int the ball Bδ(xm).

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 8/19

Implementation: 2nd order dynamical system

Two algorithms: x0 as variable:

SLIDE 30

Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 8/19

Implementation: 2nd order dynamical system

Two algorithms: xζ,0 as variable:

SLIDE 31

Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 9/19

Implementation: GA’s dynamical system

General overview:

SLIDE 32

Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 9/19

Implementation: GA’s dynamical system

Matrix representation: With Laurent Dumas (Paris VI) ith Population: Xi = {xi

l ∈ Ωad, l = 1, ..., Np}

Rewritten: Xi =     xi

1(1)

. . . xi

1(N)

. . . ... . . . xi

Np(1)

. . . xi

Np(N)

   

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 9/19

Implementation: GA’s dynamical system

■ Selection:

Xi+1/3 = SiXi

■ Crossover:

Xi+2/3 = CiXi+1/3

■ Mutation:

Xi+1 = Xi+2/3 + Ei

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 9/19

Implementation: GA’s dynamical system

The new population can be written as: Xi+1 = CiSiXi + Ei Thus genetics algorithms can be associated to:

˙ X(t) = Λ1X(t)Λ2 − X(t)

where:

X of the form: X = {xi/ i = 1, ..., Np xi ∈ Ωad}
Λi : Λi(t, X(t), P)
P are a set of fixed parameters

SLIDE 35

Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 9/19

Implementation: GA’s dynamical system

BVP formulation:      ˙ X(t) = Λ1X(t)Λ2 − X(t) X(0) = X0 mint∈[0,Z](| J(X(t)) − Jm|) < ǫ where: J(X) = min(J(xi)|xi ∈ X) Inconveniences of GA:

Slow convergence
Computational complexity
Lack of precision

Idea: Solve BVP: X0 Considered as a new variable (admissible)

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 9/19

Implementation: GA’s dynamical system

The choice of X0 is important:

SLIDE 37

Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 9/19

Implementation: GA’s dynamical system

Specific hybrid-algorithm B(X0):

X1 X1 X1 X1 S1 X2 X2 X2 X2

SLIDE 38

Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 9/19

Implementation: GA’s dynamical system

Specific hybrid-algorithm B(X0):

X2 X2 X2 X2 S2 X0

3

X0

3

X0

3

X0

3

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Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 9/19

Implementation: GA’s dynamical system

Specific hybrid-algorithm B(X0):

S3 X0

3

X0

3

X0

3

X0

3

SLIDE 40

Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 9/19

Implementation: GA’s dynamical system

Example:

SLIDE 41

Outlines

PART I: Global Optimization Methods

Problem
BVP formulation
General method for BVP

resolution

Implementation: 1st order

dynamical system

Implementation: 2nd order

dynamical system

Implementation: GA’s

dynamical system

Algorithm selection

PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 10/19

Algorithm selection

In all applications:

SD2A-2L (2-Layers structure, Steepest descent): More stable

version, simplified sensitivity.

HSGA (2-Layers structure, Genetic algorithm): No gradient

computation, coupled with steepest descent.

Classical GA: Comparison with a popular technique, coupled

with steepest descent.

SLIDE 42

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications

Problems studied
Shape Optimization of a

Microfluidic Device

Multichannel Optical Filters

Design

Portfolio Optimization Under

Constraints

Other problems

Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 11/19

PART II: Industrial Applications

SLIDE 43

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications

Problems studied
Shape Optimization of a

Microfluidic Device

Multichannel Optical Filters

Design

Portfolio Optimization Under

Constraints

Other problems

Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 12/19

Problems studied

■ Shape Optimization of a Fast-Microfluidic Mixer Device ■ Multichannel Optical Filters Design ■ Portfolio Optimization Under Constraints

SLIDE 44

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications

Problems studied
Shape Optimization of a

Microfluidic Device

Multichannel Optical Filters

Design

Portfolio Optimization Under

Constraints

Other problems

Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 13/19

Shape Optimization of a Microfluidic Device

Problem: (With David Hertzog, Juan Santiago (Stanford))

SLIDE 45

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications

Problems studied
Shape Optimization of a

Microfluidic Device

Multichannel Optical Filters

Design

Portfolio Optimization Under

Constraints

Other problems

Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 13/19

Shape Optimization of a Microfluidic Device

parameterization:

SLIDE 46

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications

Problems studied
Shape Optimization of a

Microfluidic Device

Multichannel Optical Filters

Design

Portfolio Optimization Under

Constraints

Other problems

Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 13/19

Shape Optimization of a Microfluidic Device

Modeling:

Steady equations:

C C30

90

+ Convective−Diffusion Navier−Stokes

SLIDE 47

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications

Problems studied
Shape Optimization of a

Microfluidic Device

Multichannel Optical Filters

Design

Portfolio Optimization Under

Constraints

Other problems

Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 13/19

Shape Optimization of a Microfluidic Device

Considered mesh levels: Coarse meshes : 20 secs / Fine meshes : 2 mins Computational difference: Difference of 50% !!! △ !!! Gradient difference: 10%

SLIDE 48

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications

Problems studied
Shape Optimization of a

Microfluidic Device

Multichannel Optical Filters

Design

Portfolio Optimization Under

Constraints

Other problems

Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 13/19

Shape Optimization of a Microfluidic Device

Shape optimization results:

GA: Evaluations: 5400 / Time: 7 days
HGSA: Evaluations: 2500 / Time: 3 days
SD2A-L2: Evaluations: 3400 (90 % coarse mesh) / Time: 18

hours

SLIDE 49

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications

Problems studied
Shape Optimization of a

Microfluidic Device

Multichannel Optical Filters

Design

Portfolio Optimization Under

Constraints

Other problems

Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 13/19

Shape Optimization of a Microfluidic Device

Shape optimization results:

GA: Evaluations: 5400 / Time: 7 days
HGSA: Evaluations: 2500 / Time: 3 days
SD2A-L2: Evaluations: 3400 (90 % coarse mesh) / Time: 18

hours All cases: mixing time 8µs → 1.15µs Initial mixer Optimized mixer

SLIDE 50

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications

Problems studied
Shape Optimization of a

Microfluidic Device

Multichannel Optical Filters

Design

Portfolio Optimization Under

Constraints

Other problems

Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 13/19

Shape Optimization of a Microfluidic Device

SD2A-L2 convergence history

5 10 15 20 25 1 2 3 4 5 6 7 8 Iteration µs History Best element Evolution

SLIDE 51

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications

Problems studied
Shape Optimization of a

Microfluidic Device

Multichannel Optical Filters

Design

Portfolio Optimization Under

Constraints

Other problems

Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 13/19

Shape Optimization of a Microfluidic Device

Experimental implementation ’Exp’ optimized mixer ’Num’ optimized mixer Average gain of ∼ 4µs

SLIDE 52

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications

Problems studied
Shape Optimization of a

Microfluidic Device

Multichannel Optical Filters

Design

Portfolio Optimization Under

Constraints

Other problems

Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 14/19

Multichannel Optical Filters Design

Problem proposed by Laurent Dumas (Paris VI) and Olivier Durand (Alcatel). Extended version with Yves Moreau (CEM2-UM II) ⇒ Project Math/STIC financed by CNRS 10.000 E

SLIDE 53

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications

Problems studied
Shape Optimization of a

Microfluidic Device

Multichannel Optical Filters

Design

Portfolio Optimization Under

Constraints

Other problems

Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 14/19

Multichannel Optical Filters Design

Problem proposed by Laurent Dumas (Paris VI) and Olivier Durand (Alcatel). Extended version with Yves Moreau (CEM2-UM II) ⇒ Project Math/STIC financed by CNRS 10.000 E Objectives: Inverse problems:

SLIDE 54

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications

Problems studied
Shape Optimization of a

Microfluidic Device

Multichannel Optical Filters

Design

Portfolio Optimization Under

Constraints

Other problems

Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 14/19

Multichannel Optical Filters Design

Problem proposed by Laurent Dumas (Paris VI) and Olivier Durand (Alcatel). Extended version with Yves Moreau (CEM2-UM II) ⇒ Project Math/STIC financed by CNRS 10.000 E Results: Depending on problem we obtained results with:

SD2A-L2: Best optimization technique.
HSGA
GA
Classical Sinc profile.

SLIDE 55

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications

Problems studied
Shape Optimization of a

Microfluidic Device

Multichannel Optical Filters

Design

Portfolio Optimization Under

Constraints

Other problems

Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 15/19

Portfolio Optimization Under Constraints

Problem proposed by the Asset-management team of BNP-Paribas (Guillaume Quibel, Rim Terhaoui, Sebastien Delcourt) ⇒ 5 month practice. Objectives: optimize a credit portfolio allocation structure, respecting to given constraints, in order to improve some performances (Profitability, risk measure, income) Results:

Due to modeling only HSGA is used (no sensitivity analysis).
All problems have led to portofolio improvement.
Results’ versatility has been validated by portfolio managers.

SLIDE 56

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications

Problems studied
Shape Optimization of a

Microfluidic Device

Multichannel Optical Filters

Design

Portfolio Optimization Under

Constraints

Other problems

Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 16/19

Conclusions and Perspectives

SLIDE 59

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications Conclusions and Perspectives

Conclusions and perspectives

09 Juin 2006 Soutenance de thèse - p. 18/19

Conclusions and perspectives

Conclusions:

The method is applicable and improve various optimization

methods.

The method is efficient on various industrial problems.

SLIDE 60

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications Conclusions and Perspectives

Conclusions and perspectives

09 Juin 2006 Soutenance de thèse - p. 18/19

Conclusions and perspectives

Conclusions:

The method is applicable and improve various optimization

methods.

The method is efficient on various industrial problems.

Perspectives:

For each industrial problem a deepest analysis is possible.
Explore the GA dynamical system.

SLIDE 61

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications Conclusions and Perspectives

Conclusions and perspectives

09 Juin 2006 Soutenance de thèse - p. 18/19

Conclusions and perspectives

Conclusions:

The method is applicable and improve various optimization

methods.

The method is efficient on various industrial problems.

Perspectives:

For each industrial problem a deepest analysis is possible.
Explore the GA dynamical system.

Acknowledgements: Bijan Mohammadi, Jean-Paul Dufour, Patrick Redont, Michel Cuer, Carole Duffet, Damien Isèbe (UMII), Laurent Dumas (Paris VI) Olivier Durand (Alcatel), Yves Moreau (CEM2), Juan Santiago, David Hertzog, Heinz Pitsch (Stanford), Larvi Debiane (INRIA), Alexandre Ern (ENPC), Thierry Poinsot (Cerfacs), Ramos-Del Olmo Angel Manuel (Madrid), Guillaume Quibel, Rim Terhaoui, Sebastien Delcourt (BNP-Paribas)

SLIDE 62

Outlines

PART I: Global Optimization Methods PART II: Industrial Applications Conclusions and Perspectives !!! Thank You !!! 09 Juin 2006 Soutenance de thèse - p. 19/19