A note on the c-monotonicity in optimal transport with capacity constraints

TL;DR

This paper explores the geometric properties of optimal transport solutions under capacity constraints, introducing c-capacity monotonicity as a generalization of c-cyclical monotonicity, and proves that optimal solutions satisfy this property.

Contribution

It introduces the concept of c-capacity monotonicity and proves that optimal transport plans with capacity constraints exhibit this property.

Findings

Optimal transport solutions with capacity constraints are c-capacity monotone.

c-capacity monotonicity generalizes c-cyclical monotonicity in optimal transport.

The paper provides a geometric characterization of optimal plans under capacity constraints.

Abstract

This paper studies the geometry of the optimizer for the optimal transport problem with capacity constraints. We introduce the concept of c-capacity monotonicity, which is a generalization of c-cyclical monotonicity in optimal transport. We show that the optimizer of the optimal transport problem with capacity constraints is c-capacity monotone.