Relevant bibliographies by topics / The Kantorovich duality

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  2. Dissertations / Theses
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  4. Conference papers

Academic literature on the topic 'The Kantorovich duality'

Author: Grafiati
Published: 4 June 2021
Last updated: 15 February 2022

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Journal articles on the topic "The Kantorovich duality"

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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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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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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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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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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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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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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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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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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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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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Dissertations / Theses on the topic "The Kantorovich duality"

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Oliveira, Aline Duarte de. "O teorema da dualidade de Kantorovich para o transporte de ótimo." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2011. http://hdl.handle.net/10183/32470.

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Abordaremos a teoria do transporte otimo demonstrando o teorema da dualidade de Kantorovich para uma classe ampla de funções custo. Tal resultado desempenha um papel de suma importância na teoria do transporte otimo. Uma ferramenta importante utilizada e o teorema da dualidade de Fenchel-Rockafellar, aqui enunciado e demonstrado em bastante generalidade. Demonstramos tamb em o teorema da dualidade de Kantorovich-Rubinstein, que trata do caso particular da função custo distância.
We analyze the optimal transport theory proving the Kantorovich duality theorem for a wide class of cost functions. Such result plays an extremely important role in the optimal transport theory. An important tool used here is the Fenchel-Rockafellar duality theorem, which we state and prove in a general case. We also prove the Kantorovich-Rubinstein duality theorem, which deals with the particular case of cost function given by the distance.
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Aguiar, Guilherme Ost de. "O Problema de Monge-Kantorovich para o custo quadrático." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2011. http://hdl.handle.net/10183/32384.

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Abordamos o problema do transporte otimo de Monge-Kantorovich no caso em que o custo e dado pelo quadrado da distância. Tal custo tem uma estrutura que permite a obtenção de resultados mais ricos do que o caso geral. Nosso objetivo e determinar se h a soluções para tal problema e caracteriza-las. Al em disso, tratamos informalmente do problema de transporte otimo para um custo geral.
We analyze the Monge-Kantorovich optimal transportation problem in the case where the cost function is given by the square of the Euclidean norm. Such cost has a structure which allow us to get more interesting results than the general case. Our main purpose is to determine if there are solutions to such problem and characterize them. We also give an informal treatment to the optimal transportation problem in the general case.
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Russo, Daniele. "Introduzione alla Teoria del Trasporto Ottimale e Dualità di Kantorovich." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2020. http://amslaurea.unibo.it/21788/.

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In questa tesi viene introdotta la teoria del trasporto ottimale a partire dal problema originale di Monge, che consiste nel trovare la strategia ottimale per trasportare una certa quantità di massa da uno scavo a una fortificazione. Si cerca cioè una mappa tra due spazi di probabilità che “trasporta” una misura nell’altra (quest’ultima viene detta quindi misura “push-forward”) e minimizza un funzionale detto “costo”. Dopo alcuni esempi, si introduce il caso discreto, coincidente con un problema di programmazione lineare; successivamente vengono dati alcuni risultati su funzioni semicontinue e spazi polacchi; vengono inoltre introdotte le nozioni di c-convessità e c-ciclica monotonia che permettono di enunciare e dimostrare il risultato principale della tesi: il teorema di Kantorovich, grazie al quale è possibile cercare il minimo del funzionale risolvendo un problema duale. Si danno quindi alcuni cenni di analisi convessa per poi applicare il teorema e costruire una mappa ottimale per una funzione costo quadratica e, in generale, strettamente convessa. Infine, si nota che dalla costruzione della mappa ottimale si può dedurre la cosiddetta decomposizione polare di un campo vettoriale, da cui si ricava una versione non lineare della decomposizione di Helmholtz; come ultima applicazione si risolve un problema di minimo riguardo un modello che descrive la configurazione di equilibrio di un gas utilizzando una misura “push-forward”.
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Nguyen, Van Thanh. "Problèmes de transport partiel optimal et d'appariement avec contrainte." Thesis, Limoges, 2017. http://www.theses.fr/2017LIMO0052.

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Cette thèse est consacrée à l'analyse mathématique et numérique pour les problèmes de transport partiel optimal et d'appariement avec contrainte (constrained matching problem). Ces deux problèmes présentent de nouvelles quantités inconnues, appelées parties actives. Pour le transport partiel optimal avec des coûts qui sont donnés par la distance finslerienne, nous présentons des formulations équivalentes caractérisant les parties actives, le potentiel de Kantorovich et le flot optimal. En particulier, l'EDP de condition d'optimalité permet de montrer l'unicité des parties actives. Ensuite, nous étudions en détail des approximations numériques pour lesquelles la convergence de la discrétisation et des simulations numériques sont fournies. Pour les coûts lagrangiens, nous justifions rigoureusement des caractérisations de solution ainsi que des formulations équivalentes. Des exemples numériques sont également donnés. Le reste de la thèse est consacré à l'étude du problème d'appariement optimal avec des contraintes pour le coût de la distance euclidienne. Ce problème a un comportement différent du transport partiel optimal. L'unicité de solution et des formulations équivalentes sont étudiées sous une condition géométrique. La convergence de la discrétisation et des exemples numériques sont aussi établis. Les principaux outils que nous utilisons dans la thèse sont des combinaisons des techniques d'EDP, de la théorie du transport optimal et de la théorie de dualité de Fenchel--Rockafellar. Pour le calcul numérique, nous utilisons des méthodes du lagrangien augmenté
The manuscript deals with the mathematical and numerical analysis of the optimal partial transport and optimal constrained matching problems. These two problems bring out new unknown quantities, called active submeasures. For the optimal partial transport with Finsler distance costs, we introduce equivalent formulations characterizing active submeasures, Kantorovich potential and optimal flow. In particular, the PDE of optimality condition allows to show the uniqueness of active submeasures. We then study in detail numerical approximations for which the convergence of discretization and numerical simulations are provided. For Lagrangian costs, we derive and justify rigorously characterizations of solution as well as equivalent formulations. Numerical examples are also given. The rest of the thesis presents the study of the optimal constrained matching with the Euclidean distance cost. This problem has a different behaviour compared to the partial transport. The uniqueness of solution and equivalent formulations are studied under geometric condition. The convergence of discretization and numerical examples are also indicated. The main tools which we use in the thesis are some combinations of PDE techniques, optimal transport theory and Fenchel--Rockafellar dual theory. For numerical computation, we make use of augmented Lagrangian methods
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Perrone, Paolo. "Categorical Probability and Stochastic Dominance in Metric Spaces." 2018. https://ul.qucosa.de/id/qucosa%3A32641.

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In this work we introduce some category-theoretical concepts and techniques to study probability distributions on metric spaces and ordered metric spaces. In Chapter 1 we give an overview of the concept of a probability monad, first defined by Giry. Probability monads can be interpreted as a categorical tool to talk about random elements of a space X. We can consider these random elements as formal convex combinations, or mixtures, of elements of X. Spaces where the convex combinations can be actually evaluated are called algebras of the probability monad. In Chapter 2 we define a probability monad on the category of complete metric spaces and 1-Lipschitz maps called the Kantorovich monad, extending a previous construction due to van Breugel. This monad assigns to each complete metric space X its Wasserstein space PX. It is well-known that finitely supported probability measures with rational coefficients, or empirical distributions of finite sequences, are dense in the Wasserstein space. This density property can be translated into categorical language as a colimit of a diagram involving certain powers of X. The monad structure of P, and in particular the integration map, is uniquely determined by this universal property. We prove that the algebras of the Kantorovich monad are exactly the closed convex subsets of Banach spaces. In Chapter 3 we extend the Kantorovich monad of Chapter 2 to metric spaces equipped with a partial order. The order is inherited by the Wasserstein space, and is called the stochastic order. Differently from most approaches in the literature, we define a compatibility condition of the order with the metric itself, rather then with the topology it induces. We call the spaces with this property L-ordered spaces. On L-ordered spaces, the stochastic order induced on the Wasserstein spaces satisfies itself a form of Kantorovich duality. The Kantorovich monad can be extended to the category of L-ordered metric spaces. We prove that its algebras are the closed convex subsets of ordered Banach spaces, i.e. Banach spaces equipped with a closed cone. The category of L-ordered metric spaces can be considered a 2-category, in which we can describe concave and convex maps categorically as the lax and oplax morphisms of algebras. In Chapter 4 we develop a new categorical formalism to describe operations evaluated partially. We prove that partial evaluations for the Kantorovich monad, or partial expectations, define a closed partial order on the Wasserstein space PA over every algebra A, and that the resulting ordered space is itself an algebra. We prove that, for the Kantorovich monad, these partial expectations correspond to conditional expectations in distribution. Finally, we study the relation between these partial evaluation orders and convex functions. We prove a general duality theorem extending the well-known duality between convex functions and conditional expectations to general ordered Banach spaces.
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Nguyen, Van thanh. "Problèmes de transport partiel optimal et d'appariement avec contrainte." Thesis, 2017. http://www.theses.fr/2017LIMO0052/document.

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Cette thèse est consacrée à l'analyse mathématique et numérique pour les problèmes de transport partiel optimal et d'appariement avec contrainte (constrained matching problem). Ces deux problèmes présentent de nouvelles quantités inconnues, appelées parties actives. Pour le transport partiel optimal avec des coûts qui sont donnés par la distance finslerienne, nous présentons des formulations équivalentes caractérisant les parties actives, le potentiel de Kantorovich et le flot optimal. En particulier, l'EDP de condition d'optimalité permet de montrer l'unicité des parties actives. Ensuite, nous étudions en détail des approximations numériques pour lesquelles la convergence de la discrétisation et des simulations numériques sont fournies. Pour les coûts lagrangiens, nous justifions rigoureusement des caractérisations de solution ainsi que des formulations équivalentes. Des exemples numériques sont également donnés. Le reste de la thèse est consacré à l'étude du problème d'appariement optimal avec des contraintes pour le coût de la distance euclidienne. Ce problème a un comportement différent du transport partiel optimal. L'unicité de solution et des formulations équivalentes sont étudiées sous une condition géométrique. La convergence de la discrétisation et des exemples numériques sont aussi établis. Les principaux outils que nous utilisons dans la thèse sont des combinaisons des techniques d'EDP, de la théorie du transport optimal et de la théorie de dualité de Fenchel--Rockafellar. Pour le calcul numérique, nous utilisons des méthodes du lagrangien augmenté
The manuscript deals with the mathematical and numerical analysis of the optimal partial transport and optimal constrained matching problems. These two problems bring out new unknown quantities, called active submeasures. For the optimal partial transport with Finsler distance costs, we introduce equivalent formulations characterizing active submeasures, Kantorovich potential and optimal flow. In particular, the PDE of optimality condition allows to show the uniqueness of active submeasures. We then study in detail numerical approximations for which the convergence of discretization and numerical simulations are provided. For Lagrangian costs, we derive and justify rigorously characterizations of solution as well as equivalent formulations. Numerical examples are also given. The rest of the thesis presents the study of the optimal constrained matching with the Euclidean distance cost. This problem has a different behaviour compared to the partial transport. The uniqueness of solution and equivalent formulations are studied under geometric condition. The convergence of discretization and numerical examples are also indicated. The main tools which we use in the thesis are some combinations of PDE techniques, optimal transport theory and Fenchel--Rockafellar dual theory. For numerical computation, we make use of augmented Lagrangian methods
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Book chapters on the topic "The Kantorovich duality"

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Villani, Cédric. "The Kantorovich duality." In Graduate Studies in Mathematics, 17–46. Providence, Rhode Island: American Mathematical Society, 2003. http://dx.doi.org/10.1090/gsm/058/02.

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Villani, Cédric. "Cyclical monotonicity and Kantorovich duality." In Grundlehren der mathematischen Wissenschaften, 51–92. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-540-71050-9_5.

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Levin, Vladimir L. "Abstract Convexity and the Monge-Kantorovich Duality." In Lecture Notes in Economics and Mathematical Systems, 33–72. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007. http://dx.doi.org/10.1007/978-3-540-37007-9_2.

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Lu, Xiaojun, and David Yang Gao. "Canonical Duality Method for Solving Kantorovich Mass Transfer Problem." In Advances in Mechanics and Mathematics, 105–26. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-58017-3_5.

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Gabriel-Argüelles, José Rigoberto, Martha Lorena Avendaño-Garrido, Luis Antonio Montero, and Juan González-Hernández. "Strong Duality of the Kantorovich-Rubinstein Mass Transshipment Problem in Metric Spaces." In Machine Learning, Optimization, and Data Science, 282–92. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-13709-0_24.

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Galichon, Alfred. "Monge–Kantorovich Theory." In Optimal Transport Methods in Economics. Princeton University Press, 2016. http://dx.doi.org/10.23943/princeton/9780691172767.003.0002.

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This chapter states the Monge–Kantorovich problem and provides the duality result in a fairly general setting. The primal problem is interpreted as the central planner's problem of determining the optimal assignment of workers to firms, while the dual problem is interpreted as the invisible hand's problem of obtaining a system of decentralized equilibrium prices. In general, the primal problem always has a solution (which means that an optimal assignment of workers to jobs exists), but the dual does not: the optimal assignment cannot always be decentralized by a system of prices. However, the cases where the dual problem does not have a solution are rather pathological, and in all of the cases considered in the rest of the book, both the primal and the dual problems have solutions.
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Conference papers on the topic "The Kantorovich duality"

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Dam, Nhan, Quan Hoang, Trung Le, Tu Dinh Nguyen, Hung Bui, and Dinh Phung. "Three-Player Wasserstein GAN via Amortised Duality." In Twenty-Eighth International Joint Conference on Artificial Intelligence {IJCAI-19}. California: International Joint Conferences on Artificial Intelligence Organization, 2019. http://dx.doi.org/10.24963/ijcai.2019/305.

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We propose a new formulation for learning generative adversarial networks (GANs) using optimal transport cost (the general form of Wasserstein distance) as the objective criterion to measure the dissimilarity between target distribution and learned distribution. Our formulation is based on the general form of the Kantorovich duality which is applicable to optimal transport with a wide range of cost functions that are not necessarily metric. To make optimising this duality form amenable to gradient-based methods, we employ a function that acts as an amortised optimiser for the innermost optimisation problem. Interestingly, the amortised optimiser can be viewed as a mover since it strategically shifts around data points. The resulting formulation is a sequential min-max-min game with 3 players: the generator, the critic, and the mover where the new player, the mover, attempts to fool the critic by shifting the data around. Despite involving three players, we demonstrate that our proposed formulation can be trained reasonably effectively via a simple alternative gradient learning strategy. Compared with the existing Lipschitz-constrained formulations of Wasserstein GAN on CIFAR-10, our model yields significantly better diversity scores than weight clipping and comparable performance to gradient penalty method.
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