Optimal solutions of multivariate coupling problems

Ludger Rüschendorf

Applicationes Mathematicae (1995)

  • Volume: 23, Issue: 3, page 325-338
  • ISSN: 1233-7234

Abstract

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Some necessary and some sufficient conditions are established for the explicit construction and characterization of optimal solutions of multivariate transportation (coupling) problems. The proofs are based on ideas from duality theory and nonconvex optimization theory. Applications are given to multivariate optimal coupling problems w.r.t. minimal l p -type metrics, where fairly explicit and complete characterizations of optimal transportation plans (couplings) are obtained. The results are of interest even in the one-dimensional case. For the first time an explicit criterion is given for the construction of optimal multivariate couplings for the Kantorovich metric l 1 .

How to cite

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Rüschendorf, Ludger. "Optimal solutions of multivariate coupling problems." Applicationes Mathematicae 23.3 (1995): 325-338. <http://eudml.org/doc/219135>.

@article{Rüschendorf1995,
abstract = {Some necessary and some sufficient conditions are established for the explicit construction and characterization of optimal solutions of multivariate transportation (coupling) problems. The proofs are based on ideas from duality theory and nonconvex optimization theory. Applications are given to multivariate optimal coupling problems w.r.t. minimal $l_p$-type metrics, where fairly explicit and complete characterizations of optimal transportation plans (couplings) are obtained. The results are of interest even in the one-dimensional case. For the first time an explicit criterion is given for the construction of optimal multivariate couplings for the Kantorovich metric $l_1$.},
author = {Rüschendorf, Ludger},
journal = {Applicationes Mathematicae},
keywords = {$l_p$-metric; c-convex functions; optimal couplings; transportation problem; optimal transportation; nonconvex optimization; duality theory; multivariate optimal coupling problems; explicit criterion},
language = {eng},
number = {3},
pages = {325-338},
title = {Optimal solutions of multivariate coupling problems},
url = {http://eudml.org/doc/219135},
volume = {23},
year = {1995},
}

TY - JOUR
AU - Rüschendorf, Ludger
TI - Optimal solutions of multivariate coupling problems
JO - Applicationes Mathematicae
PY - 1995
VL - 23
IS - 3
SP - 325
EP - 338
AB - Some necessary and some sufficient conditions are established for the explicit construction and characterization of optimal solutions of multivariate transportation (coupling) problems. The proofs are based on ideas from duality theory and nonconvex optimization theory. Applications are given to multivariate optimal coupling problems w.r.t. minimal $l_p$-type metrics, where fairly explicit and complete characterizations of optimal transportation plans (couplings) are obtained. The results are of interest even in the one-dimensional case. For the first time an explicit criterion is given for the construction of optimal multivariate couplings for the Kantorovich metric $l_1$.
LA - eng
KW - $l_p$-metric; c-convex functions; optimal couplings; transportation problem; optimal transportation; nonconvex optimization; duality theory; multivariate optimal coupling problems; explicit criterion
UR - http://eudml.org/doc/219135
ER -

References

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  1. J. A. Cuesta-Albertos, L. Rüschendorf and A. Tuero-Diaz (1993), Optimal coupling of multivariate distributions and stochastic processes, J. Multivariate Anal. 46, 335-361. Zbl0788.60025
  2. H. Dietrich (1988), Zur c-Konvexität und c-Subdifferenzierbarkeit von Funktionalen, Optimization 19, 355-371. 
  3. K. H. Elster und R. Nehse (1974), Zur Theorie der Polarfunktionale, ibid. 5, 3-21. Zbl0283.90049
  4. H. Kellerer (1984), Duality theorems for marginal problems, Z. Wahrsch. Verw. Gebiete 67, 399-432. Zbl0535.60002
  5. M. Knott and C. Smith (1984), On the optimal mapping of distributions, J. Optim. Theory Appl. 43, 39-49. Zbl0519.60010
  6. V. I. Levin (1992), 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, 533-548. Zbl0776.90086
  7. S. T. Rachev (1991), Probability Metric and the Stability of Stochastic Models, Wiley. Zbl0744.60004
  8. S. T. Rachev and L. Rüschendorf (1991), Solution of some transportation problems with relaxed or additional constraints, SIAM J. Control Optim., to appear. Zbl0797.60019
  9. A. W. Roberts and D. E. Varberg (1973), Convex Functions, Academic Press. Zbl0271.26009
  10. R. T. Rockafellar (1970), Convex Analysis, Princeton University Press. Zbl0193.18401
  11. L. Rüschendorf (1991a), Bounds for distributions with multivariate marginals, in: Proceedings: Stochastic Order and Decision under Risk, K. Mosler and M. Scarsini (eds.), IMS Lecture Notes 19, 285-310. Zbl0760.60019
  12. L. Rüschendorf (1991b), Fréchet-bounds and their applications, in: Advances in Probability Measures with Given Marginals, G. Dall'Aglio, S. Kotz and G. Salinetti (eds.), Kluwer Acad. Publ., 151-188. 
  13. L. Rüschendorf and S. T. Rachev (1990), A characterization of random variables with minimum L^2-distance, J. Multivariate Anal. 32, 48-54. Zbl0688.62034
  14. C. Smith and M. Knott (1992), On Hoeffding-Fréchet bounds and cyclic monotone relations, J. Multivariate Anal. 40, 328-334. Zbl0745.62055
  15. M. M. Vainberg (1973), Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, Wiley. Zbl0279.47022

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