The Experts below are selected from a list of 15402 Experts worldwide ranked by ideXlab platform
Haim Brezis - One of the best experts on this subject based on the ideXlab platform.
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remarks on the monge Kantorovich problem in the discrete setting
Comptes Rendus Mathematique, 2018Co-Authors: Haim BrezisAbstract:Abstract In Optimal Transport theory, three quantities play a central role: the minimal cost of transport, originally introduced by Monge, its relaxed version introduced by Kantorovich, and a dual formulation also due to Kantorovich. The goal of this Note is to publicize a very elementary, self-contained argument extracted from [9] , which shows that all three quantities coincide in the discrete case.
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Remarks on the Monge–Kantorovich problem in the discrete setting
Comptes Rendus Mathématique, 2018Co-Authors: Haim BrezisAbstract:In Optimal Transport theory, three quantities play a central role: the minimal cost of transport, originally introduced by Monge, its relaxed version introduced by Kantorovich, and a dual formulation also due to Kantorovich. The goal of this Note is to publicize a very elementary, self-contained argument extracted from [9], which shows that all three quantities coincide in the discrete case.
Rabee Tourky - One of the best experts on this subject based on the ideXlab platform.
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the super order dual of an ordered vector space and the riesz Kantorovich formula
Transactions of the American Mathematical Society, 2001Co-Authors: Charalambos D Aliprantis, Rabee TourkyAbstract:A classical theorem of F. Riesz and L. V. Kantorovich asserts that if L is a vector lattice and f and g are order bounded linear functionals on L, then their supremum (least upper bound) f V g exists in L ∼ and for each x ∈ L + it satisfies the so-called Riesz-Kantorovich formula: [f ∨ g](x) = sup{f(y) + g(z): y,z ∈ L + and y + z = x}. Related to the Riesz-Kantorovich formula is the following long-standing problem: If the supremum of two order bounded linear functionals f and g on an ordered vector space exists, does it then satisfy the Riesz-Kantorovich formula? In this paper, we introduce an extension of the order dual of an ordered vector space and provide some answers to this long-standing problem. The ideas regarding the Riesz-Kantorovich formula owe their origins to the study of the fundamental theorems of welfare economics and the existence of competitive equilibrium. The techniques introduced here show that the existence of decentralizing prices for efficient allocations is closely related to the abovementioned problem and to the properties of the Riesz-Kantorovich formula.
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The super order dual of an ordered vector space and the Riesz–Kantorovich formula
Transactions of the American Mathematical Society, 2001Co-Authors: Charalambos D Aliprantis, Rabee TourkyAbstract:A classical theorem of F. Riesz and L. V. Kantorovich asserts that if L is a vector lattice and f and g are order bounded linear functionals on L, then their supremum (least upper bound) f V g exists in L ∼ and for each x ∈ L + it satisfies the so-called Riesz-Kantorovich formula: [f ∨ g](x) = sup{f(y) + g(z): y,z ∈ L + and y + z = x}. Related to the Riesz-Kantorovich formula is the following long-standing problem: If the supremum of two order bounded linear functionals f and g on an ordered vector space exists, does it then satisfy the Riesz-Kantorovich formula? In this paper, we introduce an extension of the order dual of an ordered vector space and provide some answers to this long-standing problem. The ideas regarding the Riesz-Kantorovich formula owe their origins to the study of the fundamental theorems of welfare economics and the existence of competitive equilibrium. The techniques introduced here show that the existence of decentralizing prices for efficient allocations is closely related to the abovementioned problem and to the properties of the Riesz-Kantorovich formula.
Xicheng Zhang - One of the best experts on this subject based on the ideXlab platform.
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Stochastic Monge–Kantorovich problem and its duality
Stochastics, 2011Co-Authors: Xicheng ZhangAbstract:In this article, we prove the existence of a stochastic optimal transference plan for a stochastic Monge–Kantorovich problem by measurable selection theorem. A stochastic version of Kantorovich duality and the characterization of stochastic optimal transference plan are also established. Moreover, Wasserstein distance between two probability kernels is also discussed.
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Stochastic Monge-Kantorovich Problem and its Duality
arXiv: Probability, 2009Co-Authors: Xicheng ZhangAbstract:In this article we prove the existence of a stochastic optimal transference plan for a stochastic Monge-Kantorovich problem by measurable selection theorem. A stochastic version of Kantorovich duality and the characterization of stochastic optimal transference plan are also established. Moreover, Wasserstein distance between two probability kernels are discussed too.
Ioannis K. Argyros - One of the best experts on this subject based on the ideXlab platform.
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On the solution of systems of equations with constant rank derivatives
Numerical Algorithms, 2011Co-Authors: Ioannis K. Argyros, Saïd HiloutAbstract:The famous for its simplicity and clarity Newton–Kantorovich hypothesis of Newton’s method has been used for a long time as the sufficient convergence condition for solving nonlinear equations. Recently, in the elegant study by Hu et al. (J Comput Appl Math 219:110–122, 2008 ), a Kantorovich-type convergence analysis for the Gauss–Newton method (GNM) was given improving earlier results by Häubler (Numer Math 48:119–125, 1986 ), and extending some results by Argyros (Adv Nonlinear Var Inequal 8:93–99, 2005 , 2007 ) to hold for systems of equations with constant rank derivatives. In this study, we use our new idea of recurrent functions to extend the applicability of (GNM) by replacing existing conditions by weaker ones. Finally, we provide numerical examples to solve equations in cases not covered before (Häubler, Numer Math 48:119–125, 1986 ; Hu et al., J Comput Appl Math 219:110–122, 2008 ; Kontorovich and Akilov 2004 ).
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Extending the Newton-Kantorovich hypothesis for solving equations
Journal of Computational and Applied Mathematics, 2010Co-Authors: Ioannis K. Argyros, Saïd HiloutAbstract:The famous Newton-Kantorovich hypothesis (Kantorovich and Akilov, 1982 [3], Argyros, 2007 [2], Argyros and Hilout, 2009 [7]) has been used for a long time as a sufficient condition for the convergence of Newton's method to a solution of an equation in connection with the Lipschitz continuity of the Frechet-derivative of the operator involved. Here, using Lipschitz and center-Lipschitz conditions, and our new idea of recurrent functions, we show that the Newton-Kantorovich hypothesis can be weakened, under the same information. Moreover, the error bounds are tighter than the corresponding ones given by the dominating Newton-Kantorovich theorem (Argyros, 1998 [1]; [2,7]; Ezquerro and Hernandez, 2002 [11]; [3]; Proinov 2009, 2010 [16,17]). Numerical examples including a nonlinear integral equation of Chandrasekhar-type (Chandrasekhar, 1960 [9]), as well as a two boundary value problem with a Green's kernel (Argyros, 2007 [2]) are also provided in this study.
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A generalized Kantorovich theorem for nonlinear equations based on function splitting
Rendiconti del Circolo Matematico di Palermo, 2009Co-Authors: Livinus U. Uko, Ioannis K. ArgyrosAbstract:The Kantorovich theorem is a fundamental tool in nonlinear analysis for proving the existence and uniqueness of solutions of nonlinear equations arising in various fields. In the present paper we formulate and prove a generalized Kantorovich theorem that contains as special cases the Kantorovich theorem and a weak Kantorovich theorem recently proved by Uko and Argyros. An illustrative example is given to show that the new theorem is applicable in some situations in which the other two theorems are not applicable.
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A generalized Kantorovich theorem on the solvability of nonlinear equations
Aequationes Mathematicae, 2009Co-Authors: Livinus U. Uko, Ioannis K. ArgyrosAbstract:The Kantorovich theorem is a fundamental tool in nonlinear analysis for proving the existence and uniqueness of solutions of nonlinear equations arising in various fields. This theorem was weakened recently by Argyros who used a combination of Lipschitz and center-Lipschitz conditions in place of the Lipschitz conditions of the Kantorovich theorem. In the present paper we use the Argyros theorem to formulate a generalized Kantorovich theorem that enables us deduce the solvability of equations and the convergence of Newton’s method with minimal assumptions.
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A weak Kantorovich existence theorem for the solution of nonlinear equations
Journal of Mathematical Analysis and Applications, 2008Co-Authors: Livinus U. Uko, Ioannis K. ArgyrosAbstract:The Kantorovich theorem is a fundamental tool in nonlinear analysis for proving the existence and uniqueness of solutions of nonlinear equations arising in various fields. This theorem was weakened recently by Argyros who used a combination of Lipschitz and center-Lipschitz conditions in place of the Lipschitz conditions of the Kantorovich theorem. In the present paper we prove a weak Kantorovich-type theorem that gives the same conclusions as the previous two results under weaker conditions. Illustrative examples are provided in the paper.
Charalambos D Aliprantis - One of the best experts on this subject based on the ideXlab platform.
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the super order dual of an ordered vector space and the riesz Kantorovich formula
Transactions of the American Mathematical Society, 2001Co-Authors: Charalambos D Aliprantis, Rabee TourkyAbstract:A classical theorem of F. Riesz and L. V. Kantorovich asserts that if L is a vector lattice and f and g are order bounded linear functionals on L, then their supremum (least upper bound) f V g exists in L ∼ and for each x ∈ L + it satisfies the so-called Riesz-Kantorovich formula: [f ∨ g](x) = sup{f(y) + g(z): y,z ∈ L + and y + z = x}. Related to the Riesz-Kantorovich formula is the following long-standing problem: If the supremum of two order bounded linear functionals f and g on an ordered vector space exists, does it then satisfy the Riesz-Kantorovich formula? In this paper, we introduce an extension of the order dual of an ordered vector space and provide some answers to this long-standing problem. The ideas regarding the Riesz-Kantorovich formula owe their origins to the study of the fundamental theorems of welfare economics and the existence of competitive equilibrium. The techniques introduced here show that the existence of decentralizing prices for efficient allocations is closely related to the abovementioned problem and to the properties of the Riesz-Kantorovich formula.
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The super order dual of an ordered vector space and the Riesz–Kantorovich formula
Transactions of the American Mathematical Society, 2001Co-Authors: Charalambos D Aliprantis, Rabee TourkyAbstract:A classical theorem of F. Riesz and L. V. Kantorovich asserts that if L is a vector lattice and f and g are order bounded linear functionals on L, then their supremum (least upper bound) f V g exists in L ∼ and for each x ∈ L + it satisfies the so-called Riesz-Kantorovich formula: [f ∨ g](x) = sup{f(y) + g(z): y,z ∈ L + and y + z = x}. Related to the Riesz-Kantorovich formula is the following long-standing problem: If the supremum of two order bounded linear functionals f and g on an ordered vector space exists, does it then satisfy the Riesz-Kantorovich formula? In this paper, we introduce an extension of the order dual of an ordered vector space and provide some answers to this long-standing problem. The ideas regarding the Riesz-Kantorovich formula owe their origins to the study of the fundamental theorems of welfare economics and the existence of competitive equilibrium. The techniques introduced here show that the existence of decentralizing prices for efficient allocations is closely related to the abovementioned problem and to the properties of the Riesz-Kantorovich formula.
