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Notes on optimal transportation. (English) Zbl 1185.90019

This article provides a detailed introduction to the mathematical theory of optimal transportation. Based on a series of university lecture notes, it provides a useful insight in this area of optimization. The article begins with a section providing the necessary historic background of the problem. This is followed by a presentation of the general discrete optimal transportation problem and its link to the adverse selection problem, including important elements of \(u\)-convex analysis. Several theorems are presented in this section, with detailed proofs. In the fourth section, the article proceeds to study the continuous case, where the existence of a solution is non-trivial, focussing on Brenier’s theorems. The article concludes with an outline of unsolved problems in the area and a list of relevant references.

MSC:

90B06 Transportation, logistics and supply chain management
90C25 Convex programming
90C90 Applications of mathematical programming
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References:

[1] Bolton P., Dewatripont M.: Contract Theory. MIT Press, Cambridge (2005)
[2] Brenier Y.: Polar factorization and monotone rearrangements of vector functions. Comm Pure Appl Math 64, 375–417 (1991) · Zbl 0738.46011
[3] Carlier G.: A general existence result for the principal-agent problem with adverse selection. J Math Econ 35, 129–150 (2001) · Zbl 0972.91068
[4] Carlier G., Lachand-Robert T.: Regularity of solutions for some variational problems subject to a convexity constraint. Comm Pure Appl Math 54, 583–594 (2001) · Zbl 1035.49034
[5] Carlier G., Ekeland I., Touzi N.: Optimal derivatives design for mean-variance agents under adverse selection. Math Financ Econ 1, 57–80 (2007) · Zbl 1173.91379
[6] Evans, L.C., Gangbo, W.: Differential equations methods for the Monge-Kantorovitch mass transfer problem. Memoirs of the AMS 653 (1999) · Zbl 0920.49004
[7] Kantorovich L.V.: On the transfer of masses. Dokl Akad Nauk USSR 37, 113–161 (1942)
[8] Laffont J.J., Tirole J.: A Theory of Incentives in Procurement and Regulation. MIT Press, Cambridge (1993)
[9] Monge, G.: Mémoire sur la théorie des remblais et des déblais. Histoire de l’Académie des Sciences de Paris, pp. 666–704 (1781)
[10] Rochet J.C., Chone P.: Ironing, sweeping and multidimensional screening. Econometrica 66, 783–826 (1998) · Zbl 1015.91515
[11] Rochet J.C., Stole L.: The economics of multidimensional screening. Advances in economics and econometrics: theory and applications–eight world congress. In: Dewatripont, M., Hansen, L.P., Turnovsky, S.J.(eds) Econometric Society Monographs, n 36, Cambridge University Press, London (2003)
[12] Sudakov V.N.: Geometric problems in the theory of infinite dimensional probability distributions. Proc Steklov Inst 141, 1–178 (1979)
[13] Trudinger N., Wang X.-J.: On the Monge mass transfer problem. Calc Var Partial Diff Equ 13, 19–31 (2001) · Zbl 1010.49030
[14] Villani, C.: Topics in optimal transportation. Graduate Studies in Mathematics, vol. 58. AMS (2003) · Zbl 1106.90001
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