On the Monge–Kantorovich problem with additional linear constraints

By Danila Zaev. 2014

85 citations recorded by the source corpus.

The Monge–Kantorovich problem with the following additional constraint is considered: the admissible transportation plan must become zero on a fixed subspace of functions. Different subspaces give rise to different additional conditions on transportation plans. The main results are stated in general form and can be carried over to a number of important special cases. They are also valid for the Monge–Kantorovich problem whose solution is sought for the class of invariant or martingale measures. We formulate and prove a criterion for the existence of an optimal solution, a duality assertion of Kantorovich type, and a necessary geometric condition on the support of the optimal measure similar to the standard condition for c-monotonicity.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/88f5ddcac46d3b87f6eb266c2233bf946c00ec5b

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Structure of optimal martingale transport plans in general dimensions (2015, methodology): “Indeed, it is shown in [5] (see also [31]) that there exists a Borel set Λ in Rd ×Rd with π(Λ) = 1, that satisfies a certain monotonicity property, which is the martingale counterpart of the c-cyclic monotonicity that is inherent to the Monge-Kantorovich theory.”
Existence, duality, and cyclical monotonicity for weak transport costs (2018, background): “More recently, variants of this ‘monotonicity priniciple’ have been applied in transport problems for finitely or infinitely many marginals [32, 18, 26, 11, 44], the martingale version of the optimal transport problem [12, 31, 13], the Skorokhod embedding problem [9] and the distribution constrained”
Applications of weak transport theory (2020, unknown): “More recently, variants of this ‘monotonicity priniciple’ have been applied in transport problems for finitely or infinitely many marginals, the martingale version of the transport problem, the Skorokhod embedding problem, and the distribution constrained optimal stopping problem, see [43, 21, 35, 1”
Stability of martingale optimal transport and weak optimal transport (2019, unknown): “A generalization of c-cyclical monotonicity under additional linear constraints were suggested by [13, 42], which also encompass (MOT).”
Irreducible convex paving for decomposition of multidimensional martingale transport plans (2017, methodology): “Motivated by the martingale monotonicity principle of Beiglböck & Juillet [3] (see also Zaev [22] for higher dimension and general linear constraints), their strategy is to find a monotone set Γ Ă RdˆRd, where the robust superhedging holds with equality, as a support of the optimal martingale transp”
Complete Duality for Martingale Optimal Transport on the Line (2015, methodology): “4) in a fairly definitive form, generalizing and simplifying results of [7, 40].”
Causal optimal transport and its links to enlargement of filtrations and continuous-time stochastic optimization (2016, methodology): “We refer to [Zae15, BG19] for more on linearly constrained transport problems, but remark that the result below is not a consequence of theirs.”
Martingale Benamou–Brenier: A probabilistic perspective (2017, background): “idea have proved to be useful in a number of related situation, see [40, 10, 56, 29, 13, 7, 47, 14] among others.”
A new class of costs for optimal transport planning (2018, methodology): “The variational characterisation for left-curtain couplings was used to prove their stability with respect to the marginals (see [13]) and was extended for more general constraints and spaces X , Y (see [3, 20]).”