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SUMMARY:3 Novembre 2025 : Thibault Moquet & Giada Franz
DTSTART:20251103T090000Z
DTEND:20251103T110000Z
DTSTAMP:20260505T203400Z
UID:indico-event-14581@indico.math.cnrs.fr
DESCRIPTION:Thibault Moquet :\nAn Augmented Lagrangian Algorithm with inex
 act proximal method and its application to a Mean-Field Control problem\nI
 n this presentation\, we will describe our recent results on the convergen
 ce 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.\nThe Augmented Lagrangian is an algorithm for
  the minimization of a convex cost function which can be written as the su
 m of two convex functions f and g. Motivated by problems in which the dir
 ect minimization of  f+g is difficult but that of the sum of f with a re
 gularization of  g is simpler\, the Augmented Lagrangian algorithm procee
 ds by iteratively minimizing the latter sum\, modifying the regularization
  of g at each iteration. This regularization is obtained through a dual p
 rinciple\, so that our regularized problem is the dual problem associated 
 to an augmented dual cost which penalizes the distance to the dual paramet
 er μ. A more complete definition can be found for instance in [1].\nOur c
 ontribution consists here in studying the behavior of this algorithm when 
 we cannot find an exact solution of the regularized problems. More precise
 ly\, we use the Frank--Wolfe Algorithm for the resolution of these problem
 s. We show a sublinear convergence speed for this method\, in terms of the
  number of calls to the oracle for the Frank--Wolfe Algorithm.\nIn a secon
 d part\, we show that this algorithm can be used to find a numerical solut
 ion of a mean-field control problem. Our main idea consists in putting the
  contribution of the running cost and the coupling constraint in the funct
 ion f. The oracle then consists in a numerical scheme for the coupled syst
 em of PDEs Hamilton--Jacobi--Bellman and Fokker--Plank.\nReferences\n[1] J
 orge Nocedal and Stephen J. Wright\, Numerical optimization\, Springer\, 1
 999\n \nGiada Franz : \nConstruction and properties of free boundary min
 imal surfaces via min-max\nA free boundary minimal surface (FBMS) in a thr
 ee-dimensional Riemannian manifold is a critical point of the area functio
 nal 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\, Mor
 se index\, etc.). In this talk\, we will analyze these questions through 
 the lens of Simon-Smith variant of Almgren-Pitts min-max theory. More prec
 isely\, we will see how this method allows the construction of FBMS with p
 rescribed properties (symmetry\, topology\, Morse index\, etc.)\, by prese
 nting new developments and discussing the limits and perspectives of this 
 approach.\n\nhttps://indico.math.cnrs.fr/event/14581/
LOCATION:Salle Maryam Mirzakhani (201) (Institut Henri Poincaré)
URL:https://indico.math.cnrs.fr/event/14581/
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