An Iterated Projection Approach to Variational Problems Under Generalized Convexity Constraints
- 269 Downloads
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.
KeywordsPrincipal-agent problem b-convexity constraint Convexity constraint Convex envelopes Iterated projections Dykstra’s algorithm
Mathematics Subject Classification49M25 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.
- 5.Boyle, J.P., Dykstra, R.L.: A method for finding projections onto the intersection of convex sets in Hilbert spaces. In: Advances in Order Restricted Statistical Inference (Iowa City, Iowa, 1985). Lecture Notes in Statistics, vol 37, pp 28–47. Springer, Berlin, 1986Google Scholar
- 26.Mirebeau, J.-M.: Adaptive, anisotropic and hierarchical cones of discrete convex functions. arXiv preprint arXiv:1402.1561, to appear in Num. Math., 2014
- 30.Rochet, J.-C., Choné, P:. Ironing, sweeping, and multidimensional screening. Econometrica, pp. 783–826, (1998)Google Scholar