Contrainte de convexité et problème Principal Agent
le 11 fevrier 2014 – Université Paris Dauphine – Salle C 108
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10h-11h : Jean-Charles Rochet (U. Zurich – Institut fur Banking und Finance)
Stochastic Bundling
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11h-12h : Jean-Marie Mirebeau (U. Paris Dauphine -Ceremade)
Adaptive, Anisotropic and Hierarchical Cones of Convex functions. Applications to the monopolist problem.
Abstract: We address the discretization of variational problems posed on the cone of convex functions. Such problems include optimal transport with quadratic cost, as well as various long lasting geometric conjectures such as Newton’s problem of the convex body of least resistance. Our main motivation is the principal agent problem in economics, which models the impact of monopoly on product quality.Consider a two dimensional domain, sampled on a grid of N points.
We show that the cone of restrictions to the grid of convex functions is in general characterized by N^2 linear inequalities; a direct computational use of this description thus has a prohibitive complexity. We thus introduce a hierarchy of sub-cones of discrete convex functions, associated to stencils which can be adaptively, locally, and anisotropically refined. The trace on the grid of a convex function on is always contained (in an average sense over grid orientations) in a such a sub-cone defined by N ln^2 N linear constraints. Applications take advantage of these results through iterative, a-posteriori stencil refinement strategies, similar in spirit with adaptive mesh refinement methods for elliptic PDEs.
