Capacity-Constrained Optimal Transport

• Capacity-constrained optimal transport is an optimization model that imposes pointwise limits on transport plans, capturing limited resources and congestion effects.
• The formulation extends classical optimal transport by introducing a capacity density, leading to a bang–bang solution structure and ensuring uniqueness.
• Numerical methods, including double regularization and modified Sinkhorn iterations, enable efficient computation in complex, real-world scenarios.

Capacity-constrained optimal transport (COT) generalizes the classical Monge–Kantorovich transport formulation by imposing a pointwise upper bound (“capacity constraint”) on the joint transport measure. This introduces new structural, duality, and computational properties distinct from unconstrained OT and is relevant for numerous applications involving limited resource or congestion, from logistics and finance to network flows.

1. Mathematical Formulation and Primal Problem

Let $X, Y \subset \mathbb{R}^d$ be Polish, typically compact domains carrying prescribed absolutely continuous marginals $\mu \in L^1_+(X)$, $\nu \in L^1_+(Y)$, $\int_X \mu = \int_Y \nu = M < \infty$. The transport cost is specified by a measurable function $c : X \times Y \to \mathbb{R}$, often continuous and bounded. A nonnegative capacity density $\phi \in L^\infty_c(X \times Y)$, compactly supported, gives an a priori pointwise upper bound on the admissible transport density.

A coupling $h \in L^1_+(X \times Y)$ is feasible if

• $\int_Y h(x,y)\,dy = \mu(x)$ for a.e. $x$
• $\int_X h(x,y)\,dx = \nu(y)$ for a.e. $y$
• $0 \leq h(x,y) \leq \phi(x,y)$ a.e.

The admissible set is denoted $\Gamma(\mu, \nu)^\phi$. The capacity-constrained optimal transport problem seeks to

$$\min_{h \ge 0} \Bigl\{ \int_{X \times Y} c(x,y)\,h(x,y)\,dx\,dy~:~ \int_Y h(x,y)\,dy = \mu(x),~ \int_X h(x,y)\,dx = \nu(y),~ h(x,y) \le \phi(x,y) \Bigr\}$$

This modifies the classical OT linear program by the box constraint $h \leq \phi$ (Korman et al., 2012, Korman et al., 2013).

2. Structural Results: Existence, Uniqueness, and "Bang–Bang" Property

Assumptions:

• Cost: $c$ is bounded and continuous off a closed negligible set $Z \subset X \times Y$ (C1), $c \in C^2$ on $X \times Y \setminus Z$ (C2), and locally non-degenerate $\det D^2_{xy} c(x,y) \neq 0$ for all $(x,y) \notin Z$ (C3).
• Capacity: $\phi \in L^\infty(X \times Y)$, compactly supported, $\phi \geq 0$.

Existence: Under (C1) and bounded, compactly supported $\phi$, the feasible set $\Gamma(\mu, \nu)^\phi$ is weakly closed and bounded in $L^2$, and the objective is weakly continuous, so a minimizer exists.

Uniqueness & Bang–Bang Property: If also (C2–C3) and $\phi > 0$, every minimizer $h^*$ is geometrically extreme (bang–bang): there is a measurable $W \subset X \times Y$ such that $h^*(x,y) = \phi(x,y)\,1_W(x,y)$ a.e. Each minimizer is thus an extreme point of the convex feasible set—either full capacity $\phi(x,y)$ or zero at almost every location, and therefore unique (Korman et al., 2012, Korman et al., 2013).

Key ideas include a local perturbation (blow-up) at points where $0 < h(x_0, y_0) < \phi(x_0, y_0)$, showing any such “in-between” region allows a strictly lower-cost alternative, violating optimality. Any minimizer thus takes only the two values $0$ and $\phi(x,y)$, and the convexity and linearity of the setting forces uniqueness (Korman et al., 2012, Korman et al., 2013).

3. Duality Theory

The dual problem embeds the capacity constraint via a third variable $w$:

• Unconstrained Kantorovich dual: $\sup_{u+v \ge -c} \{-\int u\,d\mu -\int v\,d\nu\}$.
• Constrained dual: maximize

$$-\int_X u\,d\mu -\int_Y v\,d\nu + \int_{X \times Y} w(x,y)\,\phi(x,y)\,dx\,dy$$

over $$(u, v, w) \in L^1(X) \times L^1(Y) \times L^1(X \times Y)$$ with

$$c(x,y) + u(x) + v(y) - w(x,y) \ge 0,~~w(x,y) \le 0.$$

Complementary slackness at optimality asserts $c+u+v-w=0$ on $\operatorname{supp} h$ and $w=0$ off $\operatorname{supp} h$. The introduction of $w$ can be interpreted as a Lagrange multiplier for the upper bound, or as a “congestion penalty,” and dual certificates $(u,v,w)$ provide sensitivity analysis and lower bounds in resource allocation (Korman et al., 2012, Korman et al., 2013, Korman et al., 2013, Chen, 28 Aug 2025).

Quadratic penalization provides an alternative proof of strong duality, yielding an affine connection between primal and dual minimizers and leading to a new approach for establishing existence of dual optimizers (Korman et al., 2013, Korman et al., 2013).

4. Geometry and c-Capacity Monotonicity

The geometry of optimizers is governed by a generalization of $c$-cyclical monotonicity, called $c$-capacity monotonicity. For any finite supported probability measure $\mu$ and its competitors (same marginals, equal aggregate penalty from $w$), the optimizer minimizes the total cost over all such rearrangements. Thus, the support of any optimal plan is necessarily $c$-capacity monotone, which is a strict strengthening of cyclical monotonicity to account for the structure imposed by capacity constraints and congestion costs. In the unconstrained limit, $c$-capacity monotonicity reduces to classical $c$-cyclical monotonicity (Chen, 28 Aug 2025, Bayraktar et al., 2018).

Explicitly, the support of the optimizer lies in the “contact set” where $\varphi(x)+\psi(y)=c(x,y)+w(x,y)$, possibly filling in regions of $X \times Y$ where the capacity bound is binding, in contrast to the transport map graph in the unconstrained case.

5. Examples and Explicit Solutions

A canonical “checkerboard” example is the product $\Omega = \Lambda = [-1/2, 1/2]^d$, constant capacity $S(x,y) = p$, and $c(x,y) = -x \cdot y$. Here, symmetry and duality arguments yield an explicit optimimum $h^*(x,y) = p\,1_{x \cdot y > 0}$. Such symmetry leads to reflection relations between dual minimizers at different capacity levels and a direct determination of the bang–bang pattern (Korman et al., 2013).

Numerical simulations (e.g., for $f=g=1_{[0,1]}$, $c(x,y)=|x-y|^2$, constant $\bar h$) show that the support of $h^*$ may develop disconnected islands or singular boundary features, highlighting the complexity of the “free boundary” $\partial\{h^* = \phi\}$ (Korman et al., 2013).

6. Computational Methods

Several numerical approaches have been developed for COT:

• Double regularization (DRM): Introduces two entropic barriers corresponding to the nonnegativity and the capacity upper bound. The regularized problem admits a closed-form for $\pi_{ij}$ and reduces the feasible set to strictly positive entries, with per-iteration complexity $O(n^2)$ and memory $O(n)$. This enables substantial speedup and memory reductions compared to classical LP or iterative Bregman projection solvers (Wu et al., 2022).
• Sinkhorn-type iterations: While standard Sinkhorn algorithms do not directly extend to upper-bound constrained problems, modifications using alternating root-finding or Newton’s method on dual variables enable scalability to very large grids in both classical and network/congestion-aware settings (Wu et al., 2022, Ibrahim et al., 2023).
• ADMM and variational discretization: In the context of dynamic flows or graphs, splitting methods exploiting graph sparsity and per-edge constraints are deployed for convex variational formulations, ensuring convergence and scalability to city-scale network models (Dong et al., 1 Nov 2025).

A summary of these computational methods is provided in the table below:

Method	Main Algorithmic Idea	Complexity per Iteration
Double Reg.	Entropic barriers + Newton	$O(n^2)$ time, $O(n)$ memory
Sinkhorn-type	Alternating root-finding	$O(n^2)$
ADMM (traffic)	Minimization via splitting	$O((nk+mk)^{1.5})$, sparse

7. Applications and Extensions

Capacity-constrained OT underpins models for resource allocation with upper bounds: transport with finite vehicle/bandwidth limits, network flows with congestion, and mathematical finance with risk tolerances.

Recent extensions include:

• Ramified/branched transport: Optimal multi-paths under capacity constraints, where admissible currents are decomposed into finitely many capacity-limited components, each optimized for a concave $\alpha$-mass and satisfying tight combinatorial constraints on their structure (Xia et al., 2024, Xia et al., 12 Oct 2025).
• Dynamic/time-dependent settings: Network OT with temporal marginals, nodal-temporal flux bounds, and departure–arrival coupling, mapped to multi-marginal or unequal-dimensional OT problems and solved efficiently with path-wise Sinkhorn algorithms (Dong et al., 16 Feb 2026).
• Compression and learning: COT constraints are incorporated into adversarial optimal-transport-based point cloud compression methods, enforcing bit-rate or fidelity constraints on the learned distributions (Li et al., 2024).

Open research directions include characterizing the structure and regularity of the free boundary $\partial\{h^* = \phi\}$, analyzing higher-dimensional explicit solutions, and developing numerical methods that fully exploit the geometric and extremal “bang–bang” structure apparent in optimal capacitated plans (Korman et al., 2012, Korman et al., 2013).

---

References:

• "Optimal Transportation with Capacity Constraints" (Korman et al., 2012)
• "Insights into capacity constrained optimal transport" (Korman et al., 2013)
• "Dual potentials for capacity constrained optimal transport" (Korman et al., 2013)
• "A penalization approach to linear programming duality with application to capacity constrained transport" (Korman et al., 2013)
• "The Double Regularization Method for Capacity Constrained Optimal Transport" (Wu et al., 2022)
• "A note on the c-monotonicity in optimal transport with capacity constraints" (Chen, 28 Aug 2025)
• "Transport plans with domain constraints" (Bayraktar et al., 2018)
• "Optimization of continuous-flow over traffic networks with fundamental diagram constraints" (Dong et al., 1 Nov 2025)
• "Transport multi-paths with capacity constraints" (Xia et al., 2024)
• "Optimal transport paths with capacity induced cost function" (Xia et al., 12 Oct 2025)
• "Point Cloud Compression via Constrained Optimal Transport" (Li et al., 2024)
• "Optimal transport with constraints: from mirror descent to classical mechanics" (Ibrahim et al., 2023)
• "Temporally Flexible Transport Scheduling on Networks with Departure-Arrival Constriction and Nodal Capacity Limits" (Dong et al., 16 Feb 2026)