zbMATH — the first resource for mathematics

Optimality conditions and exact solutions of the two-dimensional Monge-Kantorovich problem. (English. Russian original) Zbl 1099.49028
J. Math. Sci., New York 133, No. 4, 1456-1463 (2006); translation from Zap. Nauchn. Semin. POMI 312, 150-164, 314 (2004).
Let \(X, Y\) be two Polish spaces, \(\sigma\) (resp. \(\tau)\) a positive Borel measure in \(X\) (resp. in \(Y)\) with \(\sigma(X) = \tau(Y),\) \(c : X \times Y \to {\mathbb R}\) a bounded continuous function (the cost function). The Monge-Kantorovich problem (1941) consists of minimizing the functional \[ \langle c, \mu \rangle = \int_{X \times Y} c(x, y) \mu(dx \,dy) \] (the cost of mass transfer) in the set of all positive Borel measures \(\mu\) in \(X \times Y\) having \(\sigma, \tau\) as marginals, which means \(\mu({\mathcal B} \times Y) = \sigma({\mathcal B})\) for every Borel set \(\mathcal B\) in \(X\) and \(\mu(X \times {\mathcal C}) = \tau({\mathcal C})\) for every Borel set \(\mathcal C\) in \(Y.\) This problem is a relaxation of the Monge problem (1781) where (in modern formulation) the functional to be minimized is \[ F(f) = \int_X c(x, f(x))\sigma(dx) \] over all measure preserving Borel maps \(f : (X, \sigma) \to (Y, \tau)\) (this means \(f(\sigma) = \tau,\) or, more precisely, \(\tau({\mathcal C}) = \sigma(f^{-1}({\mathcal C}))\) for every Borel set \(\mathcal C\) in \(Y).\) The author gives optimality conditions for both problems and explicit solutions to various two dimensional Monge and Monge-Kantorovich problems. Not much prior knowledge is assumed from the reader, numerous references are given and the emphasis is on explicit solutions, thus this paper is a good introduction to Monge and Monge-Kantorovich problems. For a paper with somewhat related aims but different contents see A. Caffarelli [Lect. Notes Math. 1813, 1–10 (2003; Zbl 1065.49027)].

49Q20 Variational problems in a geometric measure-theoretic setting
49J45 Methods involving semicontinuity and convergence; relaxation
Full Text: DOI EuDML
[1] L. Ambrosio, ”Lecture notes on optimal transport problems,” Lecture Notes in Math., 1812, 1–52 (2003). · Zbl 1047.35001
[2] L. Ambrosio, B. Kirchheim, and A. Pratelli, ”Existence of optimal transport maps for crystalline norms,” to appear. · Zbl 1076.49022
[3] Y. Brenier, ”Polar factorization and monotone rearrangement of vector-valued functions,” Comm. Pure Appl. Math., 44, 375–417 (1991). · Zbl 0738.46011 · doi:10.1002/cpa.3160440402
[4] A. M. Vershik, ”Some remarks on the infinite-dimensional linear programming problems,” Uspekhi Mat. Nauk, 25, No.5, 117–124 (1970). · Zbl 0224.90041
[5] W. Gangbo and R. J. McCann, ”The geometry of optimal transportation,” Acta Math., 177, 113–161 (1996). · Zbl 0887.49017 · doi:10.1007/BF02392620
[6] L. V. Kantorovich, ”On the translocation of masses,” Dokl. Acad. Nauk SSSR, 37, No.7–8, 199–201 (1942). · Zbl 0061.09705
[7] L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1984). · Zbl 0555.46001
[8] L. V. Kantorovich and G. S. Rubinshtein, ”On a space of completely additive functions,” Vestnik Leningrad Univ., 13, No.7, 52–59 (1958). · Zbl 0082.11001
[9] L. Caffarelli, M. Feldman, and R. J. McCann, ”Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs,” J. Amer. Math. Soc., 15, 1–26 (2002). · Zbl 1053.49032 · doi:10.1090/S0894-0347-01-00376-9
[10] V. L. Levin, ”Duality and approximation in the mass transfer problem,” in: Mathematical Economics and Functional Analysis [in Russian], Nauka, Moscow (1994), pp. 94–108.
[11] V. L. Levin, ”A formula for the optimal value in the Monge-Kantorovich problem with a smooth cost function and a characterization of cyclically monotone mappings,” Math. USSR Sb., 71, No.2, 533–548 (1992). · Zbl 0776.90086 · doi:10.1070/SM1992v071n02ABEH002136
[12] V. L. Levin, ”General Monge-Kantorovich problem and its applications in measure theory and mathematical economics,” in: Functional Analysis, Optimization, and Mathematical Economics. A Collection of Papers Dedicated to the Memory of L. V. Kantorovich, L. J. Leifman (ed.), Oxford Univ. Press, New York-Oxford (1990), pp. 141–176. · Zbl 0989.49500
[13] V. L. Levin, ”A superlinear multifunction arising in connection with mass transfer problems,” Set-Valued Anal., 4, 41–65 (1996). · Zbl 0852.47024 · doi:10.1007/BF00419373
[14] V. L. Levin, ”Reduced cost functions and their applications,” J. Math. Econom., 28, 155–186 (1997). · Zbl 0886.90048 · doi:10.1016/S0304-4068(97)00801-X
[15] V. L. Levin, ”On duality theory for non-topological variants of the mass transfer problem,” Sb. Math., 188, No.4, 571–602 (1997). · Zbl 0908.49028 · doi:10.1070/SM1997v188n04ABEH000217
[16] V. L. Levin, ”Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem,” Set-Valued Anal., 7, 7–32 (1999). · Zbl 0934.54013 · doi:10.1023/A:1008753021652
[17] V. L. Levin, ”The Monge-Kantorovich problems and stochastic preference relations,” Adv. Math. Econ., 3, 97–124 (2001). · Zbl 1019.91018
[18] V. L. Levin, ”Optimality conditions for smooth Monge solutions of the Monge-Kantorovich problem,” Funct. Anal. Appl., 36, No.2, 114–119 (2002). · Zbl 1021.49029 · doi:10.1023/A:1015666422861
[19] V. L. Levin, ”Solving the Monge and Monge-Kantorovich problems: theory and examples,” Dokl. Akad. Nauk, 67, No.1, 1–4 (2003). · Zbl 1210.49048
[20] V. L. Levin, ”Optimal solutions of the Monge problem,” Adv. Math. Econ., 6, 85–122 (2004). · Zbl 1110.49035
[21] V. L. Levin and A. A. Milyutin, ”The problem of mass transfer with a discontinuous cost function and a mass statement of the duality problem for convex extremal problems,” Russian Math. Surveys, 34, No.3, 1–78 (1979). · Zbl 0437.46064 · doi:10.1070/RM1979v034n03ABEH003996
[22] R. J. McCann, ”Exact solutions to the transportation problem on the line,” Proc. R. Soc. Lond., Ser. A, 455, 1341–1380 (1999). · Zbl 0947.90010 · doi:10.1098/rspa.1999.0364
[23] G. Monge, ”Memoire sur la theorie des deblais et de remblais,” in: Histoire de l’Academie Royale des Sciences de Paris, avec les Memoires de Mathematique et de Physique pour la meme annee, Paris (1781), pp. 666–704.
[24] A. Yu. Plakhov, ”The Newton problem on a body of the minimum average resistance,” Mat. Sb., 195, No.7, 105–126 (2004).
[25] A. Yu. Plakhov, ”Exact solutions to a one-dimensional Monge-Kantorovich problem,” Mat. Sb., 195, No.9 (2004). · Zbl 1080.49030
[26] S. T. Rachev and L. Ruschendorf, Mass Transportation Problems, Vol. 1: Theory, Vol. 2: Applications, Springer, Berlin (1998).
[27] L. Ruschendorf and S. T. Rachev, ”A characterization of random variables with minimum L 2-distance,” J. Multivariate Anal., 32, 48–54 (1990). · Zbl 0688.62034 · doi:10.1016/0047-259X(90)90070-X
[28] V. N. Sudakov, Geometric Problems in the Theory of Infinite-Dimensional Probability Distributions, Proc. Steklov Inst. Math., 141 (1979). · Zbl 0417.60029
[29] N. S. Trudinger and X. J. Wang, ”On the Monge mass transfer problem,” Calc. Var. Partial Differential Equations, 13, 19–31 (2001). · Zbl 1010.49030 · doi:10.1007/PL00009922
[30] L. Uckelmann, ”Optimal couplings between one-dimensional distributions,” in: Distributions with Given Marginals and Moment Problems, V. Benes and J. Stepan (eds.), Kluwer, Dordrecht (1997), pp. 275–281. · Zbl 0907.60022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.