Abstract
We present an algorithm to approximate the solutions to variational problems where set of admissible functions consists of convex functions. The main motivation behind the numerical method is to compute solutions to Adverse Selection problems within a Principal-Agent framework. Problems such as product lines design, optimal taxation, structured derivatives design, etc. can be studied through the scope of these models. We develop a method to estimate their optimal pricing schedules.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anguilera N.E., Morin P.: Approximating optimization problems over convex functions. Numer. Math. 111, 1–34 (2008)
Armstrong M.: Multiproduct nonlinear pricing. Econometrica 64, 51–75 (1996)
Bolton P., Dewatripoint M.: Contract Theory. MIT Press, Cambridge (2005)
Brock F., Ferone V., Kawohl B.: A symmetry problem in the calculus of variations. Calc. Var. Partial Differ. Equ. 4, 593–599 (1996)
Carlier, G.: Calculus of variations with convexity constraint. J. Nonlinear Convex Anal. 3(2), 125–143 (2002)
Carlier G.: Duality and existence for a class of mass transportation problems and economic applications. Adv. Mathe. Econ. 5, 1–21 (2003)
Carlier G., Ekeland I., Touzi N.: Optimal derivatives design for mean-variance agents under adverse selection. Math. Finan. Econ. 1, 57–80 (2007)
Carlier G., Lachand-Robert T., Maury B.: A numerical approach to variational problems subject to convexity constraints. Numer. Math. 88, 299–318 (2001)
Choné P., Le Meur H.V.J.: Non-convergence result for conformal approximation of variational problems subject to a convexity constraint. Numer. Funct. Anal. Optim. 22(5), 529–547 (2001)
Ekeland, I., Témam, R.: Convex analysis and variational problems. Classics in Applied Mathematics, vol. 28. SIAM (1976)
Evans, L.: Partial differential equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (2002)
Horst U., Moreno-Bromberg S.: Risk minimization and optimal derivative design in a principal agent game. Math. Finan. Econ. 2(1), 1–27 (2008)
Kawohl B., Schwab C.: Convergent finite elements for a class of nonconvex variational problems. IMA J. Numer. Anal. 18, 133–149 (1998)
Lachand-Robert T., Oudet É.: Minimizing within convex bodies using a convex hull method. Siam J. Optim. 16(2), 368–379 (2005)
Lachand-Robert, T., Peletier, A.: Extremal points of a functional on the set of convex functions. In: Proceedings of the American Mathematical Society, vol. 127–136, pp. 1723–1727 (1999)
Mosco U.: Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3, 510–585 (1969)
Mussa M., Rosen S.: Monopoly and product quality. J. Econ. Theory 18, 301–317 (1978)
Nicolescu, C., Persson, L.-E.: Convex Functions and their Applications. CMS Books in Mathematics. Springer Science+Business Media, Heidelberg (2006)
Rochet J.-C., Choné P.: Ironing, sweeping and multidimensional screening. Econometrica 66, 783–826 (1988)
Author information
Authors and Affiliations
Corresponding author
Additional information
We thank Guillaume Carlier and Yves Lucet for their thoughtful comments and suggestions. We would also like to acknowledge two anonymous referees and the editor, whose thorough reading and suggestions led to an improved version of our original manuscript.
Rights and permissions
About this article
Cite this article
Ekeland, I., Moreno-Bromberg, S. An algorithm for computing solutions of variational problems with global convexity constraints. Numer. Math. 115, 45–69 (2010). https://doi.org/10.1007/s00211-009-0270-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-009-0270-2