Convex Duality and Applications in PDEs and Game Theory

Tuesday 14 Feb 2023, 11:00 → 13:00 Europe/Rome

Ex-ISEF/Building-Main Lecture Hall (GSSI)

Lecturer: Filippo Santambrogio (Université Claude Bernard Lyon)

 

Title: Convex Duality and Applications in PDEs and Game Theory

 

 

Syllabus of the course:

1. Introduction to convex analysis

Convex and lsc functions. Subdifferentials, Fenchel-Legendre transforms. f**=f.

2. Duality in convex optimization

The dual of a convex optimization problem with linear constraints. Saddle points. The Uzawa algorithm. 

Fenchel-Rockafellar duality with proof of the strong duality.

3. Regularity via duality in calculus of variations and PDEs

Minimal flow problems with divergence constraints and their dual.

Sobolev regularity for the p-Laplacian, the Laplacian, and more degenerate equations.

4. The optimal transport problem

Monge and Kantorovich formulations of the OT problem. Duality.

Economic interpretations of the dual potentials as prices.

5. Wardrop equilibria

Stationary traffic problems on networks. The Braess paradox.

Relations between optimizers and equilibria via duality.

The continuous case

6. Variational Mean Field Games

An introduction to MFG. Optimal control, value function, and Hamilton-Jacobi equations.

Congestion MFG, their variational formulation and their dual. Connection with the Benamou-Brenier formula.

 

The course will be held in the Main Lecture Hall (MLH) of the ex-ISEF building, following the schedule below:

1. Tue, Feb 14: 11am - 1pm;
2. Wed, Feb 15: 11am - 1pm;
3. Wed, Feb 15, 3pm - 5pm;
4. Thurs, Feb 16: 11am - 1pm;
5. Fri, Feb 17: 3pm - 5pm;
6. Wed, Feb 22: 11am - 1pm.

For more information please consult also the calendar on the GSSI webpage.

 

 

The course lecture notes may be found at the following webpage: http://math.univ-lyon1.fr/~santambrogio/GSSI2023.html