Monge-Kantorovich interpolation with constraints and application to a parking problem

TL;DR

This work develops a theory for Monge–Kantorovich interpolation with pivot constraints, formulating the problem as $\inf_{\mu\in\mathcal A} [W_{c_0}(\mu_0,\mu)+W_{c_1}(\mu,\mu_1)]$ and exploring existence, duality, and structural properties under location and density constraints. It analyzes distance-like and strictly convex costs, establishing when minimizers concentrate on boundaries, remain interior, or become bang-bang under density caps, and proves a Γ-convergence result for penalized convex formulations. The authors introduce a parking-location model that generalizes interpolation by allowing pivot mass to be partial, relate it to a standard OT problem with a composite cost, and provide examples where parking is non-trivial. Numerical schemes based on entropic regularization and Sinkhorn-type iterations are developed to solve both constrained interpolation and parking problems, and are used to compare the two settings, illustrating how parking regions emerge under different cost regimes and constraints. Overall, the paper provides a rigorous framework for pivot-constrained transport with practical implications for urban planning (parking regions) and congestion modeling, supported by explicit 1D analyses, dual formulations, and computational experiments.

Abstract

We consider optimal transport problems where the cost for transporting a given probability measure $μ_0$ to another one $μ_1$ consists of two parts: the first one measures the transportation from $μ_0$ to an intermediate (pivot) measure $μ$ to be determined (and subject to various constraints), and the second one measures the transportation from $μ$ to $μ_1$. This leads to Monge-Kantorovich interpolation problems under constraints for which we establish various properties of the optimal pivot measures $μ$. Considering the more general situation where only some part of the mass uses the intermediate stop leads to a mathematical model for the optimal location of a parking region around a city. Numerical simulations, based on entropic regularization, are presented both for the optimal parking regions and for Monge-Kantorovich constrained interpolation problems.

Figures (7)

• Figure 1: A nonsmooth constraint set $K$ may provide a singular optimal pivot measure.
• Figure 2: The case $p=\lambda=2$. On the left $|x_0|=1$ gives $|A_{opt}|=1$; on the right $|x_0|=1/2$ gives $|A_{opt}|\simeq0.35$.
• Figure 3: The case $p=1$ and $\lambda=2$. On the left $|x_0|=1$ gives $|A_{opt}|\simeq0.97$; on the right $|x_0|=1/2$ gives $|A_{opt}|\simeq0.24$.
• Figure 4: concave cost $p=0.25$
• Figure 5: concave cost $p=0.75$

Theorems & Definitions (36)

• Example 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Remark 2.4
• Proposition 3.1
• proof
• Corollary 3.2
• proof