Math Courses

Convex Duality and Applications in PDEs and Game Theory

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

Ex-ISEF/Building-Main Lecture Hall



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: