INTERDYSCYPLINARY CENTRE
FOR MATHEMATICAL
AND COMPUTATIONAL MODELLING


A course in Convex Analysis

with plenty of solved exercises

by Alain Bossavit

Readership: Applied mathematicians, physicists.

Requisites: Calculus.

Topology, Differential calculus in normed spaces, Integration Theory, which should normally have come before, are not prerequisite strictly speaking, but for readers not drilled in that, some familiarity with epsilon-delta like analysis will help. A taste for the definition-proposition-proof style is welcome.

Motivation: The notion of "pair of convex functions in Fenchel duality" is the target. This tool is important in mathematical modelling, because it helps express a large variety of nonlinear constitutive laws, as explained in the course.

To download: Click here (about 450 k in .pdf format) or here (about 2300 k in .ps format, beware). The layout is in two vertical columns on landscape-displayed A4 pages, to make full use of a typical laptop screen.

What's in there:

1. Functions, minima, maxima

Sets, functions, relations
Real-valued functions

2. Function spaces

Affine notions
Complete spaces
Some complete functional spaces

3. Hilbert spaces

"Strong" topology
"Weak" topology

4. Convex functions, and their minimization

Semi-continuity
Convexity
Minimization

5. Separating convex sets

Separation theorems
Affine minorization of convex functions

6. Convex functions in duality

The duality relation
Conjugation, or "Fenchel transform"

Reference list

Index

This is a preliminary version, © A. Bossavit, June 03. Comments and suggestions welcome.


My address:

LGEP
11 Rue Joliot-Curie,
91192 Gif-sur-Yvette CEDEX
FRANCE

My email address: bossavit@lgep.supelec.fr

Page created June 6, 2003. Last modified June 6, 2003. © A. Bossavit