Guillaume Carlier
Paris Dauphine University
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Publication
Featured researches published by Guillaume Carlier.
Siam Journal on Mathematical Analysis | 2011
Martial Agueh; Guillaume Carlier
In this paper, we introduce a notion of barycenter in the Wasserstein space which generalizes McCanns interpolation to the case of more than two measures. We provide existence, uniqueness, characterizations, and regularity of the barycenter and relate it to the multimarginal optimal transport problem considered by Gangbo and Świech in [Comm. Pure Appl. Math., 51 (1998), pp. 23–45]. We also consider some examples and, in particular, rigorously solve the Gaussian case. We finally discuss convexity of functionals in the Wasserstein space.
SIAM Journal on Scientific Computing | 2015
Jean-David Benamou; Guillaume Carlier; Marco Cuturi; Luca Nenna; Gabriel Peyré
This article details a general numerical framework to approximate so-lutions to linear programs related to optimal transport. The general idea is to introduce an entropic regularization of the initial linear program. This regularized problem corresponds to a Kullback-Leibler Bregman di-vergence projection of a vector (representing some initial joint distribu-tion) on the polytope of constraints. We show that for many problems related to optimal transport, the set of linear constraints can be split in an intersection of a few simple constraints, for which the projections can be computed in closed form. This allows us to make use of iterative Bregman projections (when there are only equality constraints) or more generally Bregman-Dykstra iterations (when inequality constraints are in-volved). We illustrate the usefulness of this approach to several variational problems related to optimal transport: barycenters for the optimal trans-port metric, tomographic reconstruction, multi-marginal optimal trans-port and in particular its application to Breniers relaxed solutions of in-compressible Euler equations, partial un-balanced optimal transport and optimal transport with capacity constraints.
Mathematical Finance | 2010
Guillaume Carlier; Rose-Anne Dana
We consider a class of law invariant utilities, which contains the rank-dependent expected utility (RDU) and the cumulative prospect theory (CPT). We show that the computation of demand for a contingent claim when utilities are within that class, although not as simple as in the expected utility (EU) case, is still tractable. Specific attention is given to the RDU and to the CPT cases. Numerous examples are fully solved.
Journal of Mathematical Economics | 2001
Guillaume Carlier
Abstract Considering adverse selection with a continuum of types, a general characterization of implementability in terms of h -convexity is provided. This enables to write the principal’s program as a variational problem with h -convexity constraint for which existence of a solution is proved. The class of models considered here is large since the dimension of the parameter may differ from that of the contract and no structural assumption of single-crossing type is required. In particular calculus of variations problems for which admissible functions are convex ones or convex solutions to multi-time Hamilton–Jacobi equations are particular cases of the problems studied below.
Economics Papers from University Paris Dauphine | 2003
Guillaume Carlier
We establish duality, existence and uniqueness results for a class of mass transportations problems. We extend a technique of W. Gangbo [9] using the Euler Equation of the dual problem. This is done by introducing the h-Fenchel Transform and using its basic properties. The cost functions we consider satisfy a generalization of the so-called Spence-Mirrlees condition which is well-known by economists in dimension 1. We therefore end this article by a somehow unexpected application to the economic theory of incentives.
Siam Journal on Control and Optimization | 2008
Guillaume Carlier; Chloé Jimenez; Filippo Santambrogio
In the classical Monge-Kantorovich problem, the transportation cost depends only on the amount of mass sent from sources to destinations and not on the paths followed by this mass. Thus, it does not allow for congestion effects. Using the notion of traffic intensity, we propose a variant, taking into account congestion. This variant is a continuous version of a well-known traffic problem on networks that is studied both in economics and in operational research. The interest of this problem is in its relations with traffic equilibria of Wardrop type. What we prove in the paper is exactly the existence and the variational characterization of equilibria in a continuous space setting.
Journal of Optimization Theory and Applications | 2015
Jean-David Benamou; Guillaume Carlier
Many problems from mass transport can be reformulated as variational problems under a prescribed divergence constraint (static problems) or subject to a time-dependent continuity equation, which again can be formulated as a divergence constraint but in time and space. The variational class of mean field games, introduced by Lasry and Lions, may also be interpreted as a generalization of the time-dependent optimal transport problem. Following Benamou and Brenier, we show that augmented Lagrangian methods are well suited to treat such convex but non-smooth problems. They include in particular Monge historic optimal transport problem. A finite-element discretization and implementation of the method are used to provide numerical simulations and a convergence study.
Numerische Mathematik | 2001
Guillaume Carlier; Thomas Lachand-Robert; Bertrand Maury
Summary. We describe an algorithm to approximate the minimizer of an elliptic functional in the form
Journal of Economic Theory | 2012
Guillaume Carlier; Rose-Anne Dana; Alfred Galichon
\int_\Omega j(x, u, \nabla u)
Numerische Mathematik | 2016
Jean-David Benamou; Guillaume Carlier; Quentin Mérigot; Édouard Oudet
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