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TRANSPORT INEQUALITIES. A SURVEY
, 2010
"... This is a survey of recent developments in the area of transport inequalities. We investigate their consequences in terms of concentration and deviation inequalities and sketch their links with other functional inequalities and also large deviation theory. ..."
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Cited by 18 (4 self)
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This is a survey of recent developments in the area of transport inequalities. We investigate their consequences in terms of concentration and deviation inequalities and sketch their links with other functional inequalities and also large deviation theory.
A general duality theorem for the monge–kantorovich transport problem. submitted, preprint available on www.arxiv.org
 BS09] [ET99] M. Beiglböck and
, 2009
"... The duality theory of the Monge–Kantorovich transport problem is analyzed in a general setting. The spaces X, Y are assumed to be polish and equipped with Borel probability measures µ and ν. The transport cost function c: X × Y → [0, ∞] is assumed to be Borel. Our main result states that in this set ..."
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Cited by 9 (4 self)
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The duality theory of the Monge–Kantorovich transport problem is analyzed in a general setting. The spaces X, Y are assumed to be polish and equipped with Borel probability measures µ and ν. The transport cost function c: X × Y → [0, ∞] is assumed to be Borel. Our main result states that in this setting there is no duality gap, provided the optimal transport problem is formulated in a suitably relaxed way. The relaxed transport problem is defined as the limiting cost of the partial transport of masses 1−ε from (X, µ) to (Y, ν), as ε> 0 tends to zero. The classical duality theorems of H. Kellerer, where c is lower semicontinuous or uniformly bounded, quickly follow from these general results. We also show that, in the present setting, a dual optimizer always exists, provided we interpret it as a projective limit of certain finitely additive measures. Several counterexamples indicate the limitations of these results. Keywords: MongeKantorovich problem, duality, dual attainment 1
On the extremality, uniqueness and optimality of transference plans
"... Abstract. In this paper we consider the following standard problems appearing in optimal transportation theory: • when a transference plan is extremal, • when a transference plan is the unique transference plan concentrated on a set A, • when a transference plan is optimal. We show that these three ..."
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Cited by 8 (3 self)
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Abstract. In this paper we consider the following standard problems appearing in optimal transportation theory: • when a transference plan is extremal, • when a transference plan is the unique transference plan concentrated on a set A, • when a transference plan is optimal. We show that these three problems can be studied with a general approach: (1) choose some necessary conditions, depending on the problem we are considering; (2) find a partition into sets Bα where these necessary conditions become also sufficient; (3) show that all the transference plans are concentrated on ∪αBα. Explicit procedures are provided in the three cases above, the principal one being that the problem has an hidden structure of linear preorder with universally measurable graph. As by sides results, we study the disintegration theorem w.r.t. family of equivalence relations, the
A GENERALIZED DUAL MAXIMIZER FOR THE MONGEKANTOROVICH TRANSPORT PROBLEM
"... Abstract. The dual attainment of the Monge–Kantorovich transport problem is analyzed in a general setting. The spaces X, Y are assumed to be polish and equipped with Borel probability measures µ and ν. The transport cost function c: X × Y → [0, ∞] is assumed to be Borel measurable. We show that a du ..."
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Abstract. The dual attainment of the Monge–Kantorovich transport problem is analyzed in a general setting. The spaces X, Y are assumed to be polish and equipped with Borel probability measures µ and ν. The transport cost function c: X × Y → [0, ∞] is assumed to be Borel measurable. We show that a dual optimizer always exists, provided we interpret it as a projective limit of certain finitely additive measures. Our methods are functional analytic and rely on Fenchel’s perturbation technique. 1.
A GENERALIZED DUAL MAXIMIZER FOR THE MONGE–KANTOROVICH TRANSPORT PROBLEM
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Duality for increasing convex functionals with countably many marginal constraints
, 2015
"... Abstract. The main result of this paper is a convex dual representation for increasing convex functionals that are defined on a space of realvalued Borel measurable functions living on a countable product of metric spaces. Our principal assumption is that the functionals fulfill convex marginal co ..."
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Abstract. The main result of this paper is a convex dual representation for increasing convex functionals that are defined on a space of realvalued Borel measurable functions living on a countable product of metric spaces. Our principal assumption is that the functionals fulfill convex marginal constraints satisfying a tightness condition. In the special case where the marginal constraints are given by expectations or maxima of expectations, we obtain linear and sublinear versions of Kantorovich’s transport duality and the recently discovered martingale transport duality on products of countably many metric spaces.
DUALITY FOR RECTIFIED COST FUNCTIONS
"... Abstract. It is wellknown that duality in the MongeKantorovich transport problem holds true provided that the cost function c: X×Y → [0,∞] is lower semicontinuous or finitely valued, but it may fail otherwise. We present a suitable notion of rectification cr of the cost c, so that the MongeKant ..."
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Abstract. It is wellknown that duality in the MongeKantorovich transport problem holds true provided that the cost function c: X×Y → [0,∞] is lower semicontinuous or finitely valued, but it may fail otherwise. We present a suitable notion of rectification cr of the cost c, so that the MongeKantorovich duality holds true replacing c by cr. In particular, passing from c to cr only changes the value of the primal MongeKantorovich problem. Finally, the rectified function cr is lower semicontinuous as soon as X and Y are endowed with proper topologies, thus emphasizing the role of lower semicontinuity in the dualitytheory of optimal transport. 1.
A SADDLEPOINT APPROACH TO THE MONGEKANTOROVICH OPTIMAL TRANSPORT PROBLEM
"... Abstract. The MongeKantorovich problem is revisited by means of a variant of the saddlepoint method without appealing to cconjugates. A new abstract characterization of the optimal plans is obtained in the case where the cost function takes infinite values. It leads us to new explicit sufficient ..."
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Abstract. The MongeKantorovich problem is revisited by means of a variant of the saddlepoint method without appealing to cconjugates. A new abstract characterization of the optimal plans is obtained in the case where the cost function takes infinite values. It leads us to new explicit sufficient and necessary optimality conditions. As byproducts, we obtain a new proof of the wellknown Kantorovich dual equality and an improvement of the convergence of the minimizing sequences. Contents
Contents
, 710
"... Abstract. Entropy functionals (i.e. convex integral functionals) and extensions of these functionals are minimized on convex sets. This paper is aimed at reducing as much as possible the assumptions on the constraint set. Dual equalities and characterizations of ..."
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Abstract. Entropy functionals (i.e. convex integral functionals) and extensions of these functionals are minimized on convex sets. This paper is aimed at reducing as much as possible the assumptions on the constraint set. Dual equalities and characterizations of