Let and be two probability measures on the real line and let be a lower semicontinuous function on the plane. The mass transfer problem consists in determining a measure whose marginals coincide with and , and whose total cost is minimum. In this paper we present three algorithms to solve numerically this Monge-Kantorovitch problem when the commodity being shipped is one-dimensional and not necessarily confined to a bounded interval. We illustrate these numerical methods and determine the convergence rate.
@article{RO_2006__40_1_1_0, author = {Dubuc, Serge and Kagabo, Issa}, title = {Numerical solutions of the mass transfer problem}, journal = {RAIRO - Operations Research - Recherche Op\'erationnelle}, pages = {1--17}, publisher = {EDP-Sciences}, volume = {40}, number = {1}, year = {2006}, doi = {10.1051/ro:2006011}, mrnumber = {2248419}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ro:2006011/} }
TY - JOUR AU - Dubuc, Serge AU - Kagabo, Issa TI - Numerical solutions of the mass transfer problem JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2006 SP - 1 EP - 17 VL - 40 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ro:2006011/ DO - 10.1051/ro:2006011 LA - en ID - RO_2006__40_1_1_0 ER -
%0 Journal Article %A Dubuc, Serge %A Kagabo, Issa %T Numerical solutions of the mass transfer problem %J RAIRO - Operations Research - Recherche Opérationnelle %D 2006 %P 1-17 %V 40 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ro:2006011/ %R 10.1051/ro:2006011 %G en %F RO_2006__40_1_1_0
Dubuc, Serge; Kagabo, Issa. Numerical solutions of the mass transfer problem. RAIRO - Operations Research - Recherche Opérationnelle, Volume 40 (2006) no. 1, pp. 1-17. doi : 10.1051/ro:2006011. http://archive.numdam.org/articles/10.1051/ro:2006011/
[1] Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. Washington, D.C. (1964). | Zbl
and ,[2] Duality and an algorithm for a class of continuous transportation problems. Math. Oper. Res. 9 (1984) 222-231. | Zbl
and ,[3] Linear Programming in Infinite-Dimensional Spaces. Theory and Application. John Wiley & Sons, Chichester (1987). | MR | Zbl
and ,[4] Mémoire sur les déblais et les remblais des systèmes continus ou discontinus 181-208. | JFM
,[5] Le problème géométrique des déblais et remblais. Gauthier-Villars, Paris (1928). | JFM | Numdam
,[6] Déplacement de matériel continu unidimensionnel à moindre coût. RAIRO Rech. Oper., 20 (1986) 139-161. | Numdam | Zbl
and ,[7] Sur les tableaux de corrélation dont les marges sont données. Ann. Univ. Lyon 14 (1951) 53-77. | Zbl
,[8] An efficient implementation of the network simplex method. Netflow in Pisa (Pisa, 1983). Math. Program. Stud. 26 (1986) 83-111. | Zbl
,[9] The distribution of a product from several sources to numerous localities. J. Math. Phys. 20 (1941) 224-230. | JFM
,[10] Masstabinvariante Korrelations-theorie. Schr. Math. Inst. Univ. Berlin 5 (1940) 181-233. | JFM
,[11] On the translocation of masses. Doklady Akad. Nauk. SSSR 37 (1942) 199-201. | Zbl
,[12] Duality theorems for marginal problems. Z. Wahrsch. Verw. Gebiete 67 (1984) 399-432. | Zbl
,[13] The Problem of Mass Transfer with a Discontinuous Cost Function and the Mass Statement of the Duality for Convex Extremal Problems. Uspehi Mat. Nauk. 34 (1979) 3-68. | Zbl
and ,[14] Mémoire sur la théorie des déblais et des remblais. Mém. Math. Phys. Acad. Royale Sci., Paris (1781) 666-704.
,[15] Solution of some transportation problems with relaxed or additional constraints SIAM J. Control Optim. 32 (1994), 673-689. | Zbl
and ,[16] Inequalities for distributions with given marginals Ann. Prob. 8 (1980) 814-827. | Zbl
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