An iterated projection approach to variational problems under generalized convexity constraints

Guillaume Carlier (1, 2) , Xavier Dupuis (3, 1)

1 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 LUISS - Libera Università Internazionale degli Studi Sociali Guido Carli [Roma]

The principal-agent problem in economics leads to variational problems subject to global constraints of b-convexity on the admissible functions, capturing the so-called incentive-compatibility constraints. Typical examples are minimization problems subject to a convexity constraint. In a recent pathbreaking article, Fi-galli, Kim and McCann [19] identified conditions which ensure convexity of the principal-agent problem and thus raised hope on the development of numerical methods. We consider special instances of projections problems over b-convex functions and show how they can be solved numerically using Dykstra's iterated projection algorithm to handle the b-convexity constraint in the framework of [19]. Our method also turns out to be simple for convex envelope computations.

Mots clés: Principal-agent problem; b-convexity constraint; convexity constraint; convex envelopes; iterated projections; Dykstra's algorithm Mathematics subject classification: 49M25; 65K15; 90C25

Domaines: Optimisation et contrôle [math.OC]

Guillaume Carlier, Xavier Dupuis. An iterated projection approach to variational problems under generalized convexity constraints. 2015. ⟨hal-01242047⟩