Duality for rectified cost functions
01 Sep 2012-Calculus of Variations and Partial Differential Equations (Springer-Verlag)-Vol. 45, Iss: 1, pp 27-41
TL;DR: In this article, it is shown that the rectified function cr is lower semi-continuous as soon as X and Y are endowed with proper topologies, thus emphasizing the role of lower semicontinuity in the duality-theory of optimal transport.
Abstract: It is well-known that duality in the Monge–Kantorovich transport problem holds true provided that the cost function c : X × Y → [0, ∞] is lower semi-continuous or finitely valued, but it may fail otherwise. We present a suitable notion of rectification cr of the cost c, so that the Monge-Kantorovich duality holds true replacing c by cr. In particular, passing from c to cr only changes the value of the primal Monge–Kantorovich problem. Finally, the rectified function cr is lower semi-continuous as soon as X and Y are endowed with proper topologies, thus emphasizing the role of lower semi-continuity in the duality-theory of optimal transport.
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TL;DR: A quasi-sure formulation of the dual problem is introduced and shown to yield a complete duality theory for general marginals and measurable reward (cost) functions: absence of a duality gap and existence of dual optimizers as mentioned in this paper.
Abstract: We study the optimal transport between two probability measures on the real line, where the transport plans are laws of one-step martingales. A quasi-sure formulation of the dual problem is introduced and shown to yield a complete duality theory for general marginals and measurable reward (cost) functions: absence of a duality gap and existence of dual optimizers. Both properties are shown to fail in the classical formulation. As a consequence of the duality result, we obtain a general principle of cyclical monotonicity describing the geometry of optimal transports.
71 citations
TL;DR: In this article, the duality theory of the Monge-Kantorovich transport problem is analyzed in a general setting and it is shown that in this setting there is no duality gap.
Abstract: 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 $\mu$ and $
u$. The transport cost function $c:X\times Y \to [0,\infty]$ 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 $1-\varepsilon$ from $(X,\mu)$ to $(Y,
u)$, as $\varepsilon >0$ tends to zero. The classical duality theorems of H.\ Kellerer, where $c$ is lower semi-continuous or uniformly bounded, quickly follow from these general results.
42 citations
TL;DR: In this article, a causal optimal transport problem is investigated, and primal attainments and dual formulations are obtained under standard hypothesis for the related variational problems, under the standard hypothesis.
Abstract: This paper investigates causal optimal transport problems. Within this framework, primal attainments and dual formulations are obtained under standard hypothesis, for the related variational proble...
41 citations
TL;DR: In this article, the authors investigated the existence of dual optimizers in one-dimensional martingale optimal transport problems and showed that the existence holds when the cost c(x, y) is twice continuously differentiable in y.
Abstract: We investigate existence of dual optimizers in one-dimensional martingale optimal transport problems. While [Ann. Probab. 45 (2017) 3038-3074] established such existence for weak (quasi-sure) duality, [Finance Stoch. 17 (2013) 477-501] showed existence for the natural stronger (pointwise) duality may fail even in regular cases. We establish that (pointwise) dual maximizers exist when y (sic) c(x, y) is convex, or equivalent to a convex function. It follows that when marginals are compactly supported, the existence holds when the cost c(x, y) is twice continuously differentiable in y. Further, this may not be improved as we give examples with c(x, center dot) epsilon C2-epsilon, epsilon > 0, where dual attainment fails. Finally, when measures are compactly supported, we show that dual optimizers are Lipschitz if c is Lipschitz.
26 citations
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TL;DR: In this paper, the authors construct and investigate two couplings which arise as optimizers for constrained Monge-Kantorovich optimal transport problems where only supermartingales are allowed as transports.
Abstract: Two probability distributions $\mu$ and $
u$ in second stochastic order can be coupled by a supermartingale, and in fact by many. Is there a canonical choice? We construct and investigate two couplings which arise as optimizers for constrained Monge-Kantorovich optimal transport problems where only supermartingales are allowed as transports. Much like the Hoeffding-Frechet coupling of classical transport and its symmetric counterpart, the antitone coupling, these can be characterized by order-theoretic minimality properties, as simultaneous optimal transports for certain classes of reward (or cost) functions, and through no-crossing conditions on their supports; however, our two couplings have asymmetric geometries. Remarkably, supermartingale optimal transport decomposes into classical and martingale transport in several ways.
18 citations
References
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Book•
02 Jan 2013
TL;DR: In this paper, the authors provide a detailed description of the basic properties of optimal transport, including cyclical monotonicity and Kantorovich duality, and three examples of coupling techniques.
Abstract: Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.
5,524 citations
Book•
01 Mar 2003TL;DR: In this paper, the metric side of optimal transportation is considered from a differential point of view on optimal transportation, and the Kantorovich duality of the optimal transportation problem is investigated.
Abstract: Introduction The Kantorovich duality Geometry of optimal transportation Brenier's polar factorization theorem The Monge-Ampere equation Displacement interpolation and displacement convexity Geometric and Gaussian inequalities The metric side of optimal transportation A differential point of view on optimal transportation Entropy production and transportation inequalities Problems Bibliography Table of short statements Index.
4,808 citations
Book•
01 Jan 1987
TL;DR: In this article, the authors present a largely balanced approach, which combines many elements of the different traditions of the subject, and includes a wide variety of examples, exercises, and applications, in order to illustrate the general concepts and results of the theory.
Abstract: Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text attempts to present a largely balanced approach, which combines many elements of the different traditions of the subject. It includes a wide variety of examples, exercises (over 400), and applications, in order to illustrate the general concepts and results of the theory. This text provides a first basic course in classical descriptive set theory and covers material with which mathematicians interested in the subject for its own sake or those that wish to use it in their field should be familiar. Over the years, researchers in diverse areas of mathematics, such as logic and set theory, analysis, topology, probability theory, etc., have brought to the subject of descriptive set theory their own intuitions, concepts, terminology and notation.
3,340 citations
TL;DR: In this paper, Kantorovich and Akad defined a translocation of masses as a function Ψ(e, e′) defined for pairs of (B)-sets e, e − ∈ R such that: (1) it is nonnegative and absolutely additive with respect to each of its arguments, (2) Φ (e, R) = Φ(e), Ψ (R, e−∆), Ω(R, E − ∆), e− ∆ = Π(e−∀ −∆ −
Abstract: ON THE TRANSLOCATION OF MASSES L. V. Kantorovich∗ The original paper was published in Dokl. Akad. Nauk SSSR, 37, No. 7–8, 227–229 (1942). We assume that R is a compact metric space, though some of the definitions and results given below can be formulated for more general spaces. Let Φ(e) be a mass distribution, i.e., a set function such that: (1) it is defined for Borel sets, (2) it is nonnegative: Φ(e) ≥ 0, (3) it is absolutely additive: if e = e1 + e2+ · · · ; ei∩ ek = 0 (i = k), then Φ(e) = Φ(e1)+ Φ(e2) + · · · . Let Φ′(e′) be another mass distribution such that Φ(R) = Φ′(R). By definition, a translocation of masses is a function Ψ(e, e′) defined for pairs of (B)-sets e, e′ ∈ R such that: (1) it is nonnegative and absolutely additive with respect to each of its arguments, (2) Ψ(e, R) = Φ(e), Ψ(R, e′) = Φ′(e′). Let r(x, y) be a known continuous nonnegative function representing the work required to move a unit mass from x to y. We define the work required for the translocation of two given mass distributions as W (Ψ,Φ,Φ′) = ∫
1,046 citations