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Insights into capacity-constrained optimal transport.

Jonathan Korman¹, Robert J McCann

¹Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S 2E4. jkorman@math.toronto.edu

Proceedings of the National Academy of Sciences of the United States of America

|June 5, 2013

Summary

This summary is machine-generated.

Researchers found an unexpected symmetry in a variant of the optimal transportation problem. This symmetry allows for explicit solutions in multiple dimensions, building on prior work on existence and uniqueness.

Keywords:

Monge–Kantorovich mass coupling free boundary infinite dimensional linear programming resource allocation

Related Experiment Videos

Area of Science:

Mathematics
Probability and Statistics
Optimization Theory

Background:

The classical optimal transportation problem seeks the cheapest way to move 'mass' between distributions.
A variant involves finding an optimal joint measure within a set of measures sharing fixed marginals and bounded by a given density.
Prior work by Korman and McCann established the existence and uniqueness of solutions for this variant.

Purpose of the Study:

To explore an unexpected symmetry within the constrained optimal transportation problem.
To derive explicit solutions for this problem in two or more dimensions.
To connect findings with one-dimensional simulations and theoretical developments in feasible set analysis.

Main Methods:

Leveraging an identified symmetry in the optimal transportation problem variant.
Developing explicit examples inspired by one-dimensional simulations exhibiting singularities and topology.
Utilizing the identification of extreme points in the feasible set.
Employing an approach to uniqueness based on constructing feasible perturbations.

Main Results:

An unexpected symmetry was discovered in the optimal transportation problem with a density constraint.
This symmetry facilitates the derivation of explicit solutions in dimensions two and higher.
The study provides concrete examples of optimal transport plans under these specific constraints.

Conclusions:

The identified symmetry offers a powerful new tool for solving constrained optimal transportation problems.
Explicit solutions are now attainable in higher dimensions, expanding the applicability of the theory.
The findings bridge theoretical advancements with insights from numerical simulations and feasible set characterization.