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 09 Juin 2006 Soutenance de thèse - p. 1/19
Outlines • Global Optimization Methods ● Outlines -BVP formulation of optimization problems PART I: Global Optimization Methods -Implementation: 1 st and 2 nd order methods PART II: Industrial Applications -Genetic algorithms and dynamical systems Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 2/19
Outlines • Global Optimization Methods ● Outlines -BVP formulation of optimization problems PART I: Global Optimization Methods -Implementation: 1 st and 2 nd order methods PART II: Industrial Applications -Genetic algorithms and dynamical systems Conclusions and Perspectives • Industrial Applications - Shape Optimization of a Fast-Microfluidic Mixer Device - Multichannel Optical Filters Design - Portfolio Optimization Under Constraints 09 Juin 2006 Soutenance de thèse - p. 2/19
Outlines • Global Optimization Methods ● Outlines -BVP formulation of optimization problems PART I: Global Optimization Methods -Implementation: 1 st and 2 nd order methods PART II: Industrial Applications -Genetic algorithms and dynamical systems Conclusions and Perspectives • 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 09 Juin 2006 Soutenance de thèse - p. 2/19
Outlines • Global Optimization Methods ● Outlines -BVP formulation of optimization problems PART I: Global Optimization Methods -Implementation: 1 st and 2 nd order methods PART II: Industrial Applications -Genetic algorithms and dynamical systems Conclusions and Perspectives • Industrial Applications - Shape Optimization of a Fast-Microfluidic Mixer Device - Multichannel Optical Filters Design - Portfolio Optimization Under Constraints • Conclusions and perspectives 09 Juin 2006 Soutenance de thèse - p. 2/19
● Outlines PART I: Global Optimization Methods ● Problem ● BVP formulation ● General method for BVP PART I: Global Optimization Methods resolution ● Implementation: 1 st order dynamical system ● Implementation: 2 nd 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
Problem ● Outlines PART I: Global Optimization min J ( x ) Methods ● Problem x ∈ Ω ad ● BVP formulation ● General method for BVP resolution Where: ● Implementation: 1 st order dynamical system - x is the optimization parameter ● Implementation: 2 nd order dynamical system R N is the admissible space - Ω ad ∈ I ● Implementation: GA’s dynamical system ● Algorithm selection PART II: Industrial Applications Conclusions and Perspectives 09 Juin 2006 Soutenance de thèse - p. 4/19
Problem ● Outlines PART I: Global Optimization min J ( x ) Methods ● Problem x ∈ Ω ad ● BVP formulation ● General method for BVP resolution Where: ● Implementation: 1 st order dynamical system - x is the optimization parameter ● Implementation: 2 nd order dynamical system R N is the admissible space - Ω ad ∈ I ● Implementation: GA’s dynamical system ● Algorithm selection Assumptions: PART II: Industrial Applications - J ∈ C 2 (Ω ad , I Conclusions and Perspectives R) - J coercive - J m denotes: the minimum of J or a low value 09 Juin 2006 Soutenance de thèse - p. 4/19
BVP formulation Many minimization algorithms can be seen as discretizations ● Outlines of dynamical systems with initial conditions. PART I: Global Optimization Methods ● Problem ● BVP formulation ● General method for BVP resolution ● Implementation: 1 st order dynamical system ● Implementation: 2 nd 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 ● Outlines of dynamical systems with initial conditions. PART I: Global Optimization Methods ● Problem Solve numerically the optimization problem with one of those ● BVP formulation ● General method for BVP algorithms ( core optimization method ) ⇔ Solve this BVP: resolution ● Implementation: 1 st order dynamical system ● Implementation: 2 nd order dynamical system � ● Implementation: GA’s First or second order initial value problem dynamical system ● Algorithm selection min t ∈ [0 ,Z ] ( | J ( x ( t )) − J m | ) < ǫ PART II: Industrial Applications Conclusions and Perspectives where Z ∈ I R is a time and ǫ is the approximation precision. 09 Juin 2006 Soutenance de thèse - p. 5/19
BVP formulation Many minimization algorithms can be seen as discretizations ● Outlines of dynamical systems with initial conditions. PART I: Global Optimization Methods ● Problem Solve numerically the optimization problem with one of those ● BVP formulation ● General method for BVP algorithms ( core optimization method ) ⇔ Solve this BVP: resolution ● Implementation: 1 st order dynamical system ● Implementation: 2 nd order dynamical system � ● Implementation: GA’s First or second order initial value problem dynamical system ● Algorithm selection min t ∈ [0 ,Z ] ( | J ( x ( t )) − J m | ) < ǫ PART II: Industrial Applications Conclusions and Perspectives where Z ∈ I R is a time and ǫ is the approximation precision. This BVP is over-determined: more conditions than derivatives. 09 Juin 2006 Soutenance de thèse - p. 5/19
General method for BVP resolution Idea: Remove the over-determination: ● Outlines One initial condition is considered as a variable v . PART I: Global Optimization Methods ● Problem ● BVP formulation ● General method for BVP resolution ● Implementation: 1 st order dynamical system ● Implementation: 2 nd 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: ● Outlines One initial condition is considered as a variable v . PART I: Global Optimization Methods ● Problem Objective: Find a v solving BVP . ● BVP formulation ● General method for BVP resolution ● Implementation: 1 st order dynamical system ● Implementation: 2 nd 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: ● Outlines One initial condition is considered as a variable v . PART I: Global Optimization Methods ● Problem Objective: Find a v solving BVP . ● BVP formulation ● General method for BVP resolution ● Implementation: 1 st order How ? We consider a function h : Ω ad → I R : dynamical system ● Implementation: 2 nd order h ( v ) = min t ∈ [0 ,Z ] ( J ( x ( t, v )) − J m ) dynamical system ● Implementation: GA’s dynamical system ● Algorithm selection PART II: Industrial Applications Solve: Conclusions and Perspectives min h ( v ) v ∈ Ω ad 09 Juin 2006 Soutenance de thèse - p. 6/19
Implementation: 1 st order dynamical system BVP is rewritten as: ● Outlines PART I: Global Optimization Methods M ( ζ, x ( ζ )) x ζ = − d ( x ( ζ )) ● Problem ● BVP formulation x (0) = x 0 ● General method for BVP resolution min t ∈ [0 ,Z ] ( | J ( x ( t )) − J m | ) < ǫ ● Implementation: 1 st order dynamical system ● Implementation: 2 nd order Where: dynamical system ● Implementation: GA’s - x 0 ∈ Ω ad the initial condition dynamical system ● Algorithm selection - ζ is a fictitious time PART II: Industrial Applications - d a direction in Ω ad Conclusions and Perspectives - M is an operator Idea: find a x 0 solving BVP (The problem is admissible). 09 Juin 2006 Soutenance de thèse - p. 7/19
Implementation: 1 st order dynamical system Geometrical interpretation of h , using steepest descent: ● Outlines PART I: Global Optimization Methods ● Problem ● BVP formulation ● General method for BVP resolution ● Implementation: 1 st order dynamical system ● Implementation: 2 nd order dynamical system ● Implementation: GA’s dynamical system J ● Algorithm selection PART II: Industrial Applications X Conclusions and Perspectives h h(X) 09 Juin 2006 Soutenance de thèse - p. 7/19
Implementation: 1 st order dynamical system Single layer algorithm A 1 ( X 1 ) : ● Outlines PART I: Global Optimization Methods ● Problem ● BVP formulation ● General method for BVP resolution ● Implementation: 1 st order dynamical system X 1 ● Implementation: 2 nd order dynamical system ● Implementation: GA’s dynamical system J ● Algorithm selection PART II: Industrial Applications h(X 1 ) Conclusions and Perspectives h 09 Juin 2006 Soutenance de thèse - p. 7/19
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