An Iterated Projection Approach to Variational Problems Under Generalized Convexity Constraints

Abstract

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, Figalli, 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.

Ergodic control of a heterogeneous population and application to electricity pricing

“…In some specific cases (linearquadratic setting and specific agents distribution), analytic solutions can be found in one [5] or many dimensions [6], via reformulation as welfare maximization using virtual valuation technique. Extending the framework of Rochet and Choné to decomposable variational problem under convexity requirement, Carlier [7] addresses the question of the existence and uniqueness of a solution, and proposes an iterative algorithm. In the specific case R 2 , Mirebeau [8] introduces a more efficient method using an adaptive mesh based on stencils.…”

“…To apply this algorithm to the optimal design of a menu in the electricity retail market, we generalize the framework of [7] to allow for a nondecomposable (still convex) cost. Indeed, the revenue of the provider depends on the furniture cost, supposed to be an increasing function of the global consumption, see e.g.…”

“…In [19], the authors proved the convergence of the discretized problem to the continuous one, which can be extended to nondecomposable cost. Efficient numerical methods have been proposed in [7] and [8]. Let us define a regular grid Σ of X.…”