On optimal matching measures for matching problems related to the Euclidean distance

Mazón, José Manuel ; Rossi, Julio Daniel ; Toledo, Julián

Mathematica Bohemica, Tome 139 (2014) no. 4, pp. 553-566

Abstract (VO)

We deal with an optimal matching problem, that is, we want to transport two measures to a given place (the target set) where they will match, minimizing the total transport cost that in our case is given by the sum of two different multiples of the Euclidean distance that each measure is transported. We show that such a problem has a solution with an optimal matching measure supported in the target set. This result can be proved by an approximation procedure using a $p$-Laplacian system. We prove that any optimal matching measure for this problem is supported on the boundary of the target set when the two multiples that affect the Euclidean distances involved in the cost are different. Moreover, we present simple examples showing uniqueness or non-uniqueness of the optimal measure.

Détail

DOI : 10.21136/MB.2014.144132

Classification : 45G10, 49J20, 49J45, 49Q20
Keywords: mass transport; Monge-Kantorovich problem; $p$-Laplacian equation

Bibliographie

[1] Agueh, M., Carlier, G.: Barycenters in the Wasserstein space. SIAM J. Math. Anal. 43 (2011), 904-924. | DOI | MR | Zbl

[2] Ambrosio, L.: Lecture notes on optimal transport problems. Mathematical Aspects of Evolving Interfaces. Lectures given at the C.I.M.-C.I.M.E. joint Euro-summer school, Madeira, Funchal, Portugal Lecture Notes in Math. 1812 Springer, Berlin (2003), 1-52 P. Colli et al. \goodbreak. | MR | Zbl

[3] Carlier, G.: Duality and existence for a class of mass transportation problems and economic applications. S. Kusuoka et al. Advances in Mathematical Economics 5 Springer, Tokyo (2003), 1-21. | DOI | MR | Zbl

[4] Carlier, G., Ekeland, I.: Matching for teams. Econ. Theory 42 (2010), 397-418. | DOI | MR | Zbl

[5] Chiappori, P.-A., McCann, R. J., Nesheim, L. P.: Hedonic price equilibria, stable matching, and optimal transport: Equivalence, topology, and uniqueness. Econ. Theory 42 (2010), 317-354. | MR | Zbl

[6] Ekeland, I.: Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types. Econ. Theory 42 (2010), 275-315. | DOI | MR | Zbl

[7] Ekeland, I.: Notes on optimal transportation. Econ. Theory 42 (2010), 437-459. | DOI | MR | Zbl

[8] Ekeland, I.: An optimal matching problem. ESAIM, Control Optim. Calc. Var. 11 (2005), 57-71. | DOI | MR | Zbl

[9] Ekeland, I., Heckman, J. J., Nesheim, L.: Identification and Estimates of Hedonic Models. Journal of Political Economy 112 (2004), S60--S109. | DOI

[10] Evans, L. C., Gangbo, W.: Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Am. Math. Soc. 137 (1999), 66. | MR | Zbl

[11] Igbida, N., Mazón, J. M., Rossi, J. D., Toledo, J.: A Monge-Kantorovich mass transport problem for a discrete distance. J. Funct. Anal. 260 (2011), 3494-3534. | DOI | MR | Zbl

[12] Mazón, J. M., Rossi, J. D., Toledo, J.: An optimal matching problem for the Euclidean distance. SIAM J. Math. Anal. 46 (2014), 233-255. | DOI | MR | Zbl

[13] Mazón, J. M., Rossi, J. D., Toledo, J.: An optimal transportation problem with a cost given by the Euclidean distance plus import/export taxes on the boundary. Rev. Mat. Iberoam. 30 (2014), 277-308. | DOI | MR

[14] Pass, B.: Multi-marginal optimal transport and multi-agent matching problems: uniqueness and structure of solutions. Discrete Contin. Dyn. Syst. 34 (2014), 1623-1639. | DOI | MR | Zbl

[15] Pass, B.: Regularity properties of optimal transportation problems arising in hedonic pricing models. ESAIM, Control Optim. Calc. Var. 19 (2013), 668-678. | DOI | MR | Zbl

[16] Villani, C.: Optimal Transport. Old and New. Grundlehren der Mathematischen Wissenschaften 137 Springer, Berlin (2009). | DOI | MR | Zbl

[17] Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics 58 American Mathematical Society, Providence (2003). | MR | Zbl