The Double Regularization Method for Capacity Constrained Optimal Transport

TL;DR

This paper introduces a novel regularization approach for capacity constrained optimal transport, enabling more efficient and accurate solutions by simplifying the duality analysis and reducing computational complexity.

Contribution

It proposes a new regularization method that simplifies solving capacity constrained optimal transport problems and improves computational efficiency over existing algorithms.

Findings

The method satisfies capacity constraints naturally.

It converts the problem into solving single-variable equations.

Numerical experiments show superior accuracy and efficiency.

Abstract

Capacity constrained optimal transport is a variant of optimal transport, which adds extra constraints on the set of feasible couplings in the original optimal transport problem to limit the mass transported between each pair of source and sink. Based on this setting, constrained optimal transport has numerous applications, e.g., finance, network flow. However, due to the large number of constraints in this problem, existing algorithms are both time-consuming and space-consuming. In this paper, inspired by entropic regularization for the classical optimal transport problem, we introduce a novel regularization term for capacity constrained optimal transport. The regularized problem naturally satisfies the capacity constraints and consequently makes it possible to analyze the duality. Unlike the matrix-vector multiplication in the alternate iteration scheme for solving classical optimal transport, in our algorithm, each alternate iteration step is to solve several single-variable equations. Fortunately, we find that each of these equations corresponds to a single-variable monotonic function, and we convert solving these equations into finding the unique zero point of each single-variable monotonic function with Newton's method. Extensive numerical experiments demonstrate that our proposed method has a significant advantage in terms of accuracy, efficiency, and memory consumption compared with existing methods.