Convex Duality and Applications in PDEs and Game Theory
A doctoral course at GSSI, L'aquila, February 14th to 22nd, 2023.
Schedule
6 classes of 2h (approx) each, scheduled on
Feb 14, 11am-1pm
Feb 15, 11am-1pm
Feb 15, 3pm-5pm
Feb 16, 11am-1pm
Feb 17, 3pm-5pm
Feb 22, 11am-1pm
Program
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.
References and Teaching Material
Most of the material covered during the course can be found in these
notes, which have been taken frrom a book (on the calculus of
variations) in preparation. The structure into sections and chapters
of the notes does not reflect the structure of the 6 classes. The
refences inside the notes can also be
useful.
For some subjects, more complete notes can be found here:
regularity via duality
continuous Wardrop
equilibria
My book Optimal Transport for Applied Mathematicians (OTAM, see here or here for a non-official version) can also be a reference for some topics.