# Optimal solutions of multivariate coupling problems

Applicationes Mathematicae (1995)

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

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topRüschendorf, 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

top- 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
- H. Dietrich (1988), Zur c-Konvexität und c-Subdifferenzierbarkeit von Funktionalen, Optimization 19, 355-371.
- K. H. Elster und R. Nehse (1974), Zur Theorie der Polarfunktionale, ibid. 5, 3-21. Zbl0283.90049
- H. Kellerer (1984), Duality theorems for marginal problems, Z. Wahrsch. Verw. Gebiete 67, 399-432. Zbl0535.60002
- M. Knott and C. Smith (1984), On the optimal mapping of distributions, J. Optim. Theory Appl. 43, 39-49. Zbl0519.60010
- 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
- S. T. Rachev (1991), Probability Metric and the Stability of Stochastic Models, Wiley. Zbl0744.60004
- 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
- A. W. Roberts and D. E. Varberg (1973), Convex Functions, Academic Press. Zbl0271.26009
- R. T. Rockafellar (1970), Convex Analysis, Princeton University Press. Zbl0193.18401
- 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
- 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.
- 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
- C. Smith and M. Knott (1992), On Hoeffding-Fréchet bounds and cyclic monotone relations, J. Multivariate Anal. 40, 328-334. Zbl0745.62055
- M. M. Vainberg (1973), Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, Wiley. Zbl0279.47022

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