# An iterated projection approach to variational problems under generalized convexity constraints

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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 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.
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https://hal.archives-ouvertes.fr/hal-01676136
Contributor : Imb - Université de Bourgogne <>
Submitted on : Friday, January 5, 2018 - 10:59:56 AM
Last modification on : Tuesday, January 23, 2018 - 3:11:09 PM

### Citation

Guillaume Carlier, Xavier Dupuis. An iterated projection approach to variational problems under generalized convexity constraints. Applied Mathematics and Optimization, Springer Verlag (Germany), 2017, 76 (3), pp.565-592. ⟨10.1007/s00245-016-9361-5⟩. ⟨hal-01676136⟩

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