3 Novembre 2025 : Thibault Moquet & Giada Franz

lundi 3 nov. 2025, 10:00 → 12:00 Europe/Paris

Salle Maryam Mirzakhani (201) (Institut Henri Poincaré)

Thibault Moquet :

An Augmented Lagrangian Algorithm with inexact proximal method and its application to a Mean-Field Control problem

In this presentation, we will describe our recent results on the convergence of the Augmented Lagrangian Algorithm with inexact proximal method, as well as its application to the numerical approximation of a solution to a mean-field control problem.

The Augmented Lagrangian is an algorithm for the minimization of a convex cost function which can be written as the sum of two convex functions f and g. Motivated by problems in which the direct minimization of f+g is difficult but that of the sum of f with a regularization of g is simpler, the Augmented Lagrangian algorithm proceeds by iteratively minimizing the latter sum, modifying the regularization of g at each iteration. This regularization is obtained through a dual principle, so that our regularized problem is the dual problem associated to an augmented dual cost which penalizes the distance to the dual parameter μ. A more complete definition can be found for instance in [1].

Our contribution consists here in studying the behavior of this algorithm when we cannot find an exact solution of the regularized problems. More precisely, we use the Frank--Wolfe Algorithm for the resolution of these problems. We show a sublinear convergence speed for this method, in terms of the number of calls to the oracle for the Frank--Wolfe Algorithm.

In a second part, we show that this algorithm can be used to find a numerical solution of a mean-field control problem. Our main idea consists in putting the contribution of the running cost and the coupling constraint in the function f. The oracle then consists in a numerical scheme for the coupled system of PDEs Hamilton--Jacobi--Bellman and Fokker--Plank.

References

[1] Jorge Nocedal and Stephen J. Wright, Numerical optimization, Springer, 1999

 

Giada Franz : 

Construction and properties of free boundary minimal surfaces via min-max

A free boundary minimal surface (FBMS) in a three-dimensional Riemannian manifold is a critical point of the area functional with respect to variations that constrain its boundary to the boundary of the ambient manifold. It is natural to ask about the existence of FBMS (in a given ambient manifold) and their properties (topology, area, Morse index, etc.). 
In this talk, we will analyze these questions through the lens of Simon-Smith variant of Almgren-Pitts min-max theory. More precisely, we will see how this method allows the construction of FBMS with prescribed properties (symmetry, topology, Morse index, etc.), by presenting new developments and discussing the limits and perspectives of this approach.