NUMERICAL OPTIMIZATION

Laurent Dumas

Many problems occuring in industry consist in minimizing (or maximizing) a certain cost function. This course is aimed to present various optimization methods in order to solve such problems.

After a general introduction on numerical optimization, two different families of optimization methods will be presented: first, the the stochastic methods (genetic algorithms, evolution strategies, etc...) and then deterministic descent-type methods (gradient, newton, etc…). A numerical implementation with Scilab of these differents methods will be done during the computer sessions, before being applied for the resolution of an applicative problem occuring in the field of telecommunications.

Global and local optimization for real valued functions with n real parameters, with or without constraints. Examples in various applied fields

genetic algorithms, evoution strategies: general principles and examples, constraints handling, multi objective case

Part 3

Basic methods. Line search principles. Quasi Newton methods. Other methods

1. Numerical methods (6 sessions):

. Numerical implementation with Scilab of a real-valued genetic algorithm (4h30)

Application on classical test functions (Rastrigin, Griewank, etc…).

. Numerical implementation with Scilab of a first order descent method (4h30)

Application on the same previous functions

2. Applicative project: (4 sessions):

Resolution of an inverse problem in telecommunications (synthesis of a Fiber Bragg filter) by using various optimization methods (6h)

17h-18h30(course):

18h30-20h (computer session,

Implementation with Scilab of the associated direct problem (solution of a system of first order coupled ODEs)

17h-18h30(course):

18h30-20h (computer session,

Modelization of the inverse problem

Wednesday, January 25th

17h-18h30(course):

18h30-20h (computer session,

Implementation of a real coded genetic algorithms

Validation on classical test functions

17h-18h30(course):

18h30-20h (computer session,

Implementation of a real coded genetic algorithms

Validation on classical test functions

17h-18h30(course):

18h30-20h (computer session,

Implementation of a real coded genetic algorithm

Validation on classical test functions

------------------- Second week --------------------------------------------

17h-18h30(course):

18h30-20h (computer session,

Implementation of a descent method of first order

17h-18h30 (course):

18h30-20h (computer session,

Implementation of a descent method of first order

17h-18h30(course):

18h30-20h (computer session,

Constraints handling

17h-18h30(course):

18h30-20h (computer session,

Resolution of the inverse problem with a hybrid method

17h-18h30(course):

Resolution of 'real world' problems: FBG synthesis, aerodynamic shape optimization

18h30-20h (computer session,

Resolution of the inverse problem with a hybrid method.

Numercial optimization, theoretical and practical aspects: JF Bonnans, JC Gilbert, C. Lemaréchal, C. Sagastizbal, Springer Verlag 2003.

Genetic Algorithms on search, optimization and machine learning: D. Goldberg, 1989

Multi-Objective Optimization Using Evolutionary Algorithms, K. Deb, 2001

A tutorial 'How to build an avolutionary algorithm' by the EVONET community

An introduction article on 'Evolution strategies' by HG Beyer and HP Schweifel

An online course on 'Numerical Optimization' at Oxford University

An online course on 'Optimization inengineering design' at the Georgia Institute of Technology

A book Numerical recipes in C or Fortran 77 (Chapter 10 mainly)

An introduction article on 'Numerical Solution of Optimization Test-Cases by Genetic Algorithms' by N. Marco and J.A. Desideri

A short tutorial 'Introduction to Fiber Bragg Gratings'

A talk 'Optimisation of optical communication systems by means of genetic algorithms' by myself

A PhD dissertation “Synthesis and characterization of fiber Bragg gratings” by J. Skaar (chapters 2 and 3.1 mainly)

An article 'Real-coded genetic algorithm for Bragg grating parameter synthesis' by G. Cormier and R. Boudreau

An article 'Multi-objective and constrained design of fibre Bragg gratings using evolutionary algorithms' by S. Manos and L. Poladian

An article "Semi deterministic vs Genetic Algorithms for global optimization of multichannel optical filters" by myself, B. Ivorra, B. Mohammadi, P. Redont and O. Durand

Ohers articles on Lennard-Jones problem: article 1 article 2, article 3, article 4 thesis

A program for the computation of a FBG reflectivity spectrum: directFBG.sci

A real valued genetic algorithm: GA.sci

A steepest descent method with backtracking linesearch: steepest.sci

A program for solving the Lennard Jones problem (LJ.sci)

Here are 16 pictures taken during my stay at UP: UP-photos.zip (zip file, 10Mo)

The subject (.doc file) is here

The documents must be sent at dumas@ann.jussieu.fr.

You can also use this email in case of technical problems or misunderstandings during the project writing.

Sorry for the delay...