Applied Mathematics & Optimization

, Volume 76, Issue 3, pp 565–592 | Cite as

An Iterated Projection Approach to Variational Problems Under Generalized Convexity Constraints

  • Guillaume CarlierEmail author
  • Xavier Dupuis


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.


Principal-agent problem b-convexity constraint Convexity constraint Convex envelopes Iterated projections  Dykstra’s algorithm 

Mathematics Subject Classification

49M25 65K15 90C25 



The authors benefited from the hospitality of the Fields Institute (Toronto, Canada), where part of the present research was conducted during the Thematic Semester on Variational Problems in Physics, Economics and Geometry. They gratefully acknowledge support from the ANR, through the projects ISOTACE (ANR-12-MONU-0013), OPTIFORM (ANR-12-BS01-0007) and from INRIA through the “action exploratoire” MOKAPLAN and wish to thank J.-D. Benamou for stimulating discussions. G.C. gratefully acknowledges the hospitality of the Mathematics and Statistics Department at UVIC (Victoria, Canada) and support from the CNRS.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.CEREMADE, UMR CNRS 7534Université Paris IX DauphineParis Cedex 16France
  2. 2.Department of Economics and FinanceLUISS Guido CarliRomeItaly

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