Journal ArticleDOI

# Dual potentials for capacity constrained optimal transport

01 Sep 2015-Calculus of Variations and Partial Differential Equations (Springer Berlin Heidelberg)-Vol. 54, Iss: 1, pp 573-584

AbstractOptimal transportation with capacity constraints, a variant of the well-known optimal transportation problem, is concerned with transporting one probability density $$f \in L^1(\mathbf {R}^m)$$ onto another one $$g \in L^1(\mathbf {R}^n)$$ so as to optimize a cost function $$c \in L^1(\mathbf {R}^{m+n})$$ while respecting the capacity constraints $$0\le h \le \bar{h}\in L^\infty (\mathbf {R}^{m+n})$$ . A linear programming duality for this problem was first proposed by Levin. In this note, we prove under mild assumptions on the given data, the existence of a pair of $$L^1$$ -functions optimizing the dual problem. Using these functions, which can be viewed as Lagrange multipliers to the marginal constraints $$f$$ and $$g$$ , we characterize the solution $$h$$ of the primal problem. We expect these potentials to play a key role in any further analysis of $$h$$ . Moreover, starting from Levin’s duality, we derive the classical Kantorovich duality for unconstrained optimal transport. In tandem with results obtained in our companion paper [Korman et al. J Convex Anal arXiv:1309.3022 [8] (in press)], this amounts to a new and elementary proof of Kantorovich’s duality.

Topics:

Content maybe subject to copyright    Report

##### Citations
More filters

01 Jan 2014
Abstract: An approach to linear programming duality is proposed which relies on quadratic penalization, so that the relation between solutions to the penalized primal and dual problems becomes ane. This yields a new proof of Levin’s duality theorem for capacity-constrained optimal transport as an innite-dimension al application.

6 citations

### Cites background or result from "Dual potentials for capacity constr..."

• ...Combining the techniques presented in the following with some of the results derived by the authors in a companion paper [7], we also provide a new elementary proof of Kantorovich’s duality....

[...]

• ...In [7], the authors prove (under some additional assumptions) that the supremum on the right is attained by triple of functions....

[...]

Journal ArticleDOI
Abstract: The Kantorovich optimal transport problemwith a density constraint onmeasures on an infinite-dimensional space is considered. In this setting, the admissible transport plan is nonnegative and majorized by a given constraint function. The existence and the uniqueness of a solution of this problem are proved.

5 citations

Posted Content
Abstract: A new approach to linear programming duality is proposed which relies on quadratic penalization, so that the relation between solutions to the penalized primal and dual problems becomes ane. This yields a new proof of Levin’s duality theorem for capacity-constrained optimal transport as an innite-dimension al application.

3 citations

Posted Content
Abstract: This paper mainly addresses the Monge mass transfer problem in the 1-D case. Through an ingenious approximation mechanism, one transforms the Monge problem into a sequence of minimization problems, which can be converted into a sequence of nonlinear differential equations with constraints by variational method. The existence and uniqueness of the solution for each equation can be demonstrated by applying the canonical duality method. Moreover, the duality method gives a sequence of perfect dual maximization problems. In the final analysis, one constructs the approximation of optimal mapping for the Monge problem according to the theoretical results.

2 citations

Journal ArticleDOI
Abstract: We consider optimal transportation with constraint, as did Korman and McCann (2013, 2015), provide simplifications and generalizations of their examples and results, and provide some new examples and results.

1 citations

##### References
More filters

Book
01 Jan 1973

14,517 citations

Book
02 Jan 2013
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.

4,558 citations

Journal ArticleDOI
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′) = ∫

839 citations

### "Dual potentials for capacity constr..." refers background in this paper

• ...As a consequence of our analysis, we are also able to deduce Kantorovich’s duality theorem for the unconstrained optimal transportation problem as a singular limit h̄→ ∞....

[...]

• ...In the optimal transport problem of Monge [10] and Kantorovich [2], one is given distributions f(x) of sources and g(y) of sinks, and is asked which pairing h(x, y) ≥ 0 of sources with sinks minimizes a given transportation cost c(x, y)....

[...]

• ...The duality theory initiated by Kantorovich provides a key tool for the analysis of this question....

[...]

• ...The remainder of this paper is organized as follows: In Section 2, we derive weak duality and complementary slackness conditions; Section 3 contains the key (coercivity) estimates; Section 4 contains the main result; in Section 5, we give a new elementary proof of the Kantorovich duality; we finally conclude this paper with a discussion on future work in Section 6....

[...]

• ...In tandem with the results obtained in the companion paper [7], this amounts to a new and elementary proof of the Kantorovich duality theorem....

[...]

BookDOI
01 Jan 1998
Abstract: Modifications of the Monge-Kantorovich Problems: Transportation Problems with Relaxed or Additional Constraints.- Application of Kantorovich-Type Metrics to Various Probabilistic-Type Limit Theorems.- Mass Transportation Problems and Recursive Stochastic Equations.- Stochastic Differential Equations and Empirical Measures.

791 citations

### "Dual potentials for capacity constr..." refers background in this paper

• ...14 of [11], for which a new proof is given in [7]), it has not been clear until now whether the dual problem admits solutions....

[...]

• ...14 of [11] to a handwritten manuscript of Levin....

[...]

Posted Content
Abstract: In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x,y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing in particular on recent developments in the regularity theory for Monge-Ampere type equations. An application to microeconomics will also be described, which amounts to finding the equilibrium price distribution for a monopolist marketing a multidimensional line of products to a population of anonymous agents whose preferences are known only statistically.

69 citations

### "Dual potentials for capacity constr..." refers methods in this paper

• ...However, the compactification techniques used to find them in the unconstrained problem [9] [13] fail miserably when h̄ 6= +∞....

[...]

• ...As in the theory for the unconstrained Monge-Kantorovich problem [9] [13] which has developed since the work of Brenier [1], we expect our characterization of primal optimizers using dual solutions will be the starting point for any future analysis of their analytic or geometric properties....

[...]