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Journal articles on the topic 'The Kantorovich duality'

Author: Grafiati

Published: 4 June 2021

Last updated: 15 February 2022

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1

Ramachandran, Doraiswamy, Doraiswamy Ramachandran, Ludger Ruschendorf, and Ludger Ruschendorf. "On the Monge - Kantorovich duality theorem." Teoriya Veroyatnostei i ee Primeneniya 45, no. 2 (2000): 403–9. http://dx.doi.org/10.4213/tvp474.

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2

Zhang, Xicheng. "Stochastic Monge–Kantorovich problem and its duality." Stochastics 85, no. 1 (November 17, 2011): 71–84. http://dx.doi.org/10.1080/17442508.2011.624627.

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3

Edwards, D. A. "A simple proof in Monge–Kantorovich duality theory." Studia Mathematica 200, no. 1 (2010): 67–77. http://dx.doi.org/10.4064/sm200-1-4.

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4

Levin, V. L. "Best approximation problems relating to Monge-Kantorovich duality." Sbornik: Mathematics 197, no. 9 (October 31, 2006): 1353–64. http://dx.doi.org/10.1070/sm2006v197n09abeh003802.

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5

Gozlan, Nathael, Cyril Roberto, Paul-Marie Samson, and Prasad Tetali. "Kantorovich duality for general transport costs and applications." Journal of Functional Analysis 273, no. 11 (December 2017): 3327–405. http://dx.doi.org/10.1016/j.jfa.2017.08.015.

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6

Olubummo, Yewande. "On duality for a generalized Monge–Kantorovich problem." Journal of Functional Analysis 207, no. 2 (February 2004): 253–63. http://dx.doi.org/10.1016/j.jfa.2003.10.006.

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7

Daryaei, M. H., and A. R. Doagooei. "Topical functions: Hermite-Hadamard type inequalities and Kantorovich duality." Mathematical Inequalities & Applications, no. 3 (2018): 779–93. http://dx.doi.org/10.7153/mia-2018-21-56.

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8

CHEN, YONGXIN, WILFRID GANGBO, TRYPHON T. GEORGIOU, and ALLEN TANNENBAUM. "On the matrix Monge–Kantorovich problem." European Journal of Applied Mathematics 31, no. 4 (August 5, 2019): 574–600. http://dx.doi.org/10.1017/s0956792519000172.

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The classical Monge–Kantorovich (MK) problem as originally posed is concerned with how best to move a pile of soil or rubble to an excavation or fill with the least amount of work relative to some cost function. When the cost is given by the square of the Euclidean distance, one can define a metric on densities called the Wasserstein distance. In this note, we formulate a natural matrix counterpart of the MK problem for positive-definite density matrices. We prove a number of results about this metric including showing that it can be formulated as a convex optimisation problem, strong duality, an analogue of the Poincaré–Wirtinger inequality and a Lax–Hopf–Oleinik–type result.

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9

BOUSCH, THIERRY. "La distance de réarrangement, duale de la fonctionnelle de Bowen." Ergodic Theory and Dynamical Systems 32, no. 3 (April 5, 2011): 845–68. http://dx.doi.org/10.1017/s014338571000088x.

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AbstractOn the space of signed invariant measures of Aℕ, one constructs a norm (and hence a distance) that seems to have a particular significance in dynamics. I shall present some of its properties, in particular a duality theorem à la Kantorovich–Rubinshtein, which gives an expression of this distance using couplings.

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10

Mikami, Toshio. "A simple proof of duality theorem for Monge-Kantorovich problem." Kodai Mathematical Journal 29, no. 1 (March 2006): 1–4. http://dx.doi.org/10.2996/kmj/1143122381.

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11

Beiglböck, Mathias, Christian Léonard, and Walter Schachermayer. "A general duality theorem for the Monge–Kantorovich transport problem." Studia Mathematica 209, no. 2 (2012): 151–67. http://dx.doi.org/10.4064/sm209-2-4.

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12

Hernández-Lerma, Onésimo, and J. Rigoberto Gabriel. "Strong duality of the Monge-Kantorovich mass transfer problem in metric spaces." Mathematische Zeitschrift 239, no. 3 (March 1, 2002): 579–91. http://dx.doi.org/10.1007/s002090100325.

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13

Mengue, Jairo K., and Elismar R. Oliveira. "Duality results for iterated function systems with a general family of branches." Stochastics and Dynamics 17, no. 03 (March 26, 2017): 1750021. http://dx.doi.org/10.1142/s0219493717500216.

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Given [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] compact metric spaces, we consider two iterated function systems [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are contractions. Let [Formula: see text] be the set of probabilities [Formula: see text] with [Formula: see text]-marginal being holonomic with respect to [Formula: see text] and [Formula: see text]-marginal being holonomic with respect to [Formula: see text]. Given [Formula: see text] and [Formula: see text], let [Formula: see text] be the set of probabilities in [Formula: see text] having [Formula: see text]-marginal [Formula: see text] and [Formula: see text]-marginal [Formula: see text]. Let [Formula: see text] be the relative entropy of [Formula: see text] with respect to [Formula: see text] and [Formula: see text] be the relative entropy of [Formula: see text] with respect to [Formula: see text]. Given a cost function [Formula: see text], let [Formula: see text]. We will prove the duality equation: [Formula: see text] In particular, if [Formula: see text] and [Formula: see text] are single points and we drop the entropy, the equation above can be rewritten as the Kantorovich duality for the compact spaces [Formula: see text] and a continuous cost function [Formula: see text].

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14

ALIBERT, J. J., G. BOUCHITTÉ, and T. CHAMPION. "A new class of costs for optimal transport planning." European Journal of Applied Mathematics 30, no. 6 (November 29, 2018): 1229–63. http://dx.doi.org/10.1017/s0956792518000669.

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We study a class of optimal transport planning problems where the reference cost involves a non-linear function G(x, p) representing the transport cost between the Dirac measure δx and a target probability p. This allows to consider interesting models which favour multi-valued transport maps in contrast with the classical linear case ($G(x,p)=\int c(x,y)dp$) where finding single-valued optimal transport is a key issue. We present an existence result and a general duality principle which apply to many examples. Moreover, under a suitable subadditivity condition, we derive a Kantorovich–Rubinstein version of the dual problem allowing to show existence in some regular cases. We also consider the well studied case of Martingale transport and present some new perspectives for the existence of dual solutions in connection with Γ-convergence theory.

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15

POWELL, S. "Kantorovich's hidden duality." IMA Journal of Management Mathematics 8, no. 3 (1997): 195–201. http://dx.doi.org/10.1093/imaman/8.3.195.

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16

Ramachandran, D., and L. Rüschendorf. "On the Monge-Kantorovitch Duality Theorem." Theory of Probability & Its Applications 45, no. 2 (January 2001): 350–56. http://dx.doi.org/10.1137/s0040585x97978300.

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17

Chung, Nhan-Phu, and Thanh-Son Trinh. "Unbalanced optimal total variation transport problems and generalized Wasserstein barycenters." Proceedings of the Royal Society of Edinburgh: Section A Mathematics, June 4, 2021, 1–27. http://dx.doi.org/10.1017/prm.2021.27.

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In this paper, we establish a Kantorovich duality for unbalanced optimal total variation transport problems. As consequences, we recover a version of duality formula for partial optimal transports established by Caffarelli and McCann; and we also get another proof of Kantorovich–Rubinstein theorem for generalized Wasserstein distance $\widetilde {W}_1^{a,b}$ proved before by Piccoli and Rossi. Then we apply our duality formula to study generalized Wasserstein barycenters. We show the existence of these barycenters for measures with compact supports. Finally, we prove the consistency of our barycenters.

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18

Ciosmak, Krzysztof J. "Optimal transport of vector measures." Calculus of Variations and Partial Differential Equations 60, no. 6 (September 19, 2021). http://dx.doi.org/10.1007/s00526-021-02095-2.

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AbstractWe develop and study a theory of optimal transport for vector measures. We resolve in the negative a conjecture of Klartag, that given a vector measure on Euclidean space with total mass zero, the mass of any transport set is again zero. We provide a counterexample to the conjecture. We generalise the Kantorovich–Rubinstein duality to the vector measures setting. Employing the generalisation, we answer the conjecture in the affirmative provided there exists an optimal transport with absolutely continuous marginals of its total variation.

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