An Iterated Projection Approach to Variational Problems Under Generalized Convexity Constraints

• Published in: Applied mathematics & optimization Vol. 76; no. 3; pp. 565 - 592
• Main Authors: Carlier, Guillaume, Dupuis, Xavier
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2017; Springer Nature B.V; Springer Verlag (Germany)
• Subjects: Calculus of Variations and Optimal Control; Optimization; Control; Convexity; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical and Computational Physics; Numerical methods; Simulation; Systems Theory; Theoretical; 65K15; Convex envelopes; convexity constraint; 90C25; Principal-agent problem; Iterated projections; 49M25; Dykstra’s algorithm; b-convexity constraint; Convexity constraint
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-016-9361-5

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 et al. (J Econ Theory 146(2):454–478, 2011 ) 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 (Figalli et al. in J Econ Theory 146(2):454–478, 2011 ). Our method also turns out to be simple for convex envelope computations.