Le prochain séminaire de l’équipe Mokaplan aura lieu le mercredi 23 janvier à 10h30 à l’INRIA Paris (2 rue Simone Iff) en salle Jacques-Louis Lions 1.
Nous aurons le plaisir d’écouter Pierre WEISS (CNRS/ ITAV) et Luca NENNA (Université Paris-Sud).
Titre: A convergence analysis of some exchange algorithms
Exchange algorithms are a class of optimization methods to solve semi-infinite programs: optimization problems over a finite dimensional variable with an infinite number of constraints. These problems arise naturally in a variety of situations. We will focus here on their use for the resolution of inverse problems without discretization of the underlying domain. A typical example is super-resolution by using a total variation regularizer. Our main result states the linear convergence rate of the method under technical assumptions such as a non degeneracy condition.
Titre: Unequal Dimensional Optimal Transport, Monge-Ampère equations and beyond
This talk is devoted to variational problems on the set of probability
measures which involve optimal transport between unequal dimensional
spaces. In particular, we study the minimization of a functional consisting
of the sum of a term reflecting the cost of (unequal dimensional) optimal
transport between one fixed and one free marginal, and another functional
of the free marginal (of various forms). Motivating applications include
Cournot-Nash equilibria where the strategy space is lower dimensional
than the space of agent types. For a variety of different forms of the term
described above, we show that a nestedness condition, which is known to
yield much improved tractability of the optimal transport problem, holds
for any minimizer. Depending on the exact form of the functional, we
exploit this to find Monge-Ampère type equations characterising solutions,
prove convergence of an iterative scheme to compute the solution, and
prove regularity results. This a joint work with Brendan Pass.