Guillaume Carlier

Barycenters in the Wasserstein Space

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.

Iterative Bregman Projections for Regularized Transportation Problems

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.

OPTIMAL DEMAND FOR CONTINGENT CLAIMS WHEN AGENTS HAVE LAW INVARIANT UTILITIES

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.

A general existence result for the principal-agent problem with adverse selection

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.

Duality and existence for a class of mass transportation problems and economic applications

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.

Optimal Transportation with Traffic Congestion and Wardrop Equilibria

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.

Augmented Lagrangian Methods for Transport Optimization, Mean Field Games and Degenerate Elliptic Equations

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.

A numerical approach to variational problems subject to convexity constraint

Summary. We describe an algorithm to approximate the minimizer of an elliptic functional in the form

Pareto efficiency for the concave order and multivariate comonotonicity

\int_\Omega j(x, u, \nabla u)

Discretization of functionals involving the Monge---Ampère operator

on the set