Capacity-Constrained Optimal Transport

• Capacity-constrained optimal transport is a framework that imposes pointwise bounds on mass transfers, leading to bang–bang solutions with unique geometric properties.
• The dual formulation introduces potential functions and slack variables that enforce complementary slackness and optimality conditions under capacity limits.
• Explicit examples and computational algorithms demonstrate the framework's practical relevance in network flow, supply chain management, and statistical treatment assignments.

The optimal transport problem with capacity constraints is a variant of the classical Monge–Kantorovich optimal transport framework in which the joint transportation plan is restricted by pointwise upper bounds, modeling scenarios where limited resources or physical, operational, or regulatory caps preclude arbitrary mass transfer between source and target distributions. This introduces combinatorial and analytic complexity, but also arises naturally in applications such as network flow, supply chain management, and statistical treatment assignment under resource limits. Recent advances have established existence, uniqueness, duality foundations, geometric structure, and computational strategies for these problems, and have revealed deep connections to convex analysis, infinite-dimensional linear programming, and regularity theory.

1. Mathematical Formulation and Model Structure

The classical optimal transport problem seeks a joint density $h$ (or a coupling) with given marginals $f$ and $g$ that minimizes

$$I_c(h) = \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} c(x, y) h(x, y)\, dx\, dy,$$

where $c(x, y)$ is the cost and $h$ belongs to the set of admissible couplings $$\Gamma(f, g) = \{ h \geq 0: (h)_x = f, (h)_y = g \}$$. The capacity-constrained variant imposes an additional measurable, bounded, compactly supported capacity bound $\bar{h}(x, y)$: $$\Gamma(f, g)^{\bar{h}} = \{ h \in L^1(\mathbb{R}^{2d}): h \geq 0,\ h(x, y) \leq \bar{h}(x, y) \ \text{a.e.},\ (h)_x = f,\ (h)_y = g \}.$$ The capacity-constrained optimal transport problem is then

$$\min_{h \in \Gamma(f, g)^{\bar{h}}} \int c(x, y) h(x, y) dx\, dy.$$

This formulation models the case where each origin-destination pair $(x, y)$ can transport at most $\bar{h}(x, y)$, reflecting pointwise constraints on transport density.

2. Uniqueness, Bang–Bang Structure, and Geometric Characterization

A fundamental result is the uniqueness of optimal solutions under mild regularity assumptions on the cost—specifically, continuity and local non-degeneracy ($\det D_{xy}^2 c(x, y) \ne 0$ a.e.)—and bounded, compactly supported $\bar{h}$ (Korman et al., 2012). For such data, every optimizer is “geometrically extreme” or “bang–bang”: there exists a measurable set $W \subset \mathbb{R}^{2d}$ such that

$$h^*(x, y) = \bar{h}(x, y)\cdot 1_W(x, y)\ \text{a.e.}$$

Thus, at almost every point, the optimal plan either saturates the capacity (transfers at the maximum allowed rate), or transports nothing; no intermediate values occur. This property excludes convex combinations of distinct minimizers unless they coincide, enforcing uniqueness.

This bang–bang or extremal structure admits explicit descriptions in examples, such as the “checkerboard” case on $[-1/2, 1/2]^2$ with quadratic cost and constant capacities, where

$$h^*(x, y) = \begin{cases} 2 & \text{if } x, y \in [-1/2, 0] \text{ or } x, y \in [0, 1/2], \ 0 & \text{otherwise}. \end{cases}$$

In (Korman et al., 2013), the set of extreme points of the feasible set is characterized: $h$ is extreme if and only if $h = 1_W \cdot \bar{h}$ for some measurable $W$, thus every optimizer and every extreme point is of bang–bang type. This simplifies geometric analysis and has broad implications for uniqueness and explicit computation.

3. Duality Theory and Optimality Conditions

The duality structure generalizes Kantorovich duality by integrating the effect of capacity constraints (Korman et al., 2012, Korman et al., 2013, Korman et al., 2013). The dual problem is: $$\sup_{(u, v, w)} \left\{ -\int u(x) f(x) dx - \int v(y) g(y) dy + \int w(x, y)\, \bar{h}(x, y) dx\, dy \right\},$$ with

$$c(x, y) + u(x) + v(y) - w(x, y) \geq 0,\quad w(x, y) \leq 0.$$

Here, $u$ and $v$ serve as potentials enforcing marginal constraints, while $w$ is associated with saturation of the capacity constraint.

A candidate plan $h$ is optimal if there exist dual potentials such that

$$\begin{aligned} &c(x, y) + u(x) + v(y) - w(x, y) = 0\quad &\text{on the active set } \{h(x, y) = \bar{h}(x, y)\},\ &c(x, y) + u(x) + v(y) - w(x, y) > 0\quad &\text{if } h(x, y) = 0,\ &w(x, y) = 0\quad &\text{if } h(x, y) < \bar{h}(x, y). \end{aligned}$$

Complementary slackness thus classifies regions in $(x, y)$ according to whether the transport is saturated or zero.

Quadratic penalization methods (Korman et al., 2013) provide regularization and computational tractability, yielding affine relations between primal and dual optimizers and furnishing an alternative, elementary proof of duality. In the limit of infinite capacity ($\bar{h} \to +\infty$), the dual reduces to the classical Kantorovich dual formulation (Korman et al., 2013).

4. Geometry: Monotonicity, Structure of the Optimizer, and Extensions

The geometry of the optimizer is informed by monotonicity principles. In the unconstrained case, optimal couplings concentrate on $c$-cyclically monotone sets. For capacity-constrained problems, the notion generalizes to $c$-capacity monotonicity (Chen, 28 Aug 2025): a set $T \subset X \times Y$ is $c$-capacity monotone if for every finitely supported discrete measure supported in $T$, and for any competitor measure with matching marginals and total “additional cost” (arising from the saturated dual variable $w(x, y)$), the original configuration does not admit a lower total cost.

The optimizer is then concentrated on a $c$-capacity monotone set, guaranteeing a geometric optimality property even in the presence of pointwise constraints. This generalization evidences the robustness of monotonic structures in optimal transport theory under strong feasibility restrictions.

Further, in domains with domain constraints more general than just pointwise capacity (e.g., limitations on support, martingale constraints, or bounded volatility), the geometry may be characterized via monotonicity principles or dual variational representations, as in (Bayraktar et al., 2018).

5. Explicit Examples and Symmetries

Explicit examples and symmetries arise in special cases—when marginals and capacities are uniform or the cost admits particular algebraic symmetry. In (Korman et al., 2013), a symmetric “checkerboard” solution is constructed for quadratic and linear costs, with regions of saturated transport arranged according to the sign of $x\cdot y$. The general machinery includes Holder-duality symmetries between problems indexed by dual exponents, and geometric reflection properties of the optimizer.

Moreover, these results provide some of the first explicit multidimensional solutions to the capacity-constrained optimal transport problem, demonstrating that the optimal plan may exhibit singularities, non-smooth boundaries, and intricate geometric structure induced by the intersection of the feasibility and optimality domains.

6. Extensions and Applications

Applications of capacity-constrained optimal transport arise wherever mass (or resource) flux is limited locally or globally. Network flow, supply chains, traffic assignment, and service/resource allocation (e.g., in healthcare, where hospital or facility capacities are fixed) all fit into this paradigm (Daoud et al., 2022, Wu et al., 2022). The theory extends naturally to time-dependent (dynamic) and path-constrained settings (Kerrache et al., 2022, Bauer et al., 24 Feb 2024), where both density and momentum (flux) are bounded, and to branched transportation with limits on “branch” weights (Xia et al., 11 Feb 2024), as well as to time-sequenced multi-period problems (Shi, 14 Feb 2025).

On the computational side, algorithmic innovations leverage the structure of the dual (Newton methods, entropic or double regularization (Wu et al., 2022), and scaling algorithms) to efficiently solve high-dimensional or semi-discrete problems, often reducing complexity by exploiting sparsity and decoupling variables through duality or iterative projections (Bansil et al., 2019).

In statistical decision theory, the optimal assignment of treatments under resource constraints can be recast as a capacity-constrained optimal transport problem; this permits decision-theoretic analysis of treatment assignment rules under nontrivial resource limits (Sunada et al., 13 Jun 2025).

7. Broader Perspectives and Future Directions

The integration of capacity constraints into optimal transport stabilizes and regularizes the solution structure, yielding uniqueness, geometric extremality, and robust characterizations that persist under generalizations. Duality theory continues to unify the constrained and unconstrained frameworks, with penalization and variational approaches bridging to computational practice. The geometric insights (such as $c$-capacity monotonicity) both deepen analytic understanding and support solution verification in high-dimensional and applied settings.

Ongoing research addresses regularity of dual potentials, stability of the optimizer under perturbations, algorithmic acceleration for large-scale and high-dimensional instances, and extension to further classes of constraints (e.g., dynamic, path-dependent, or stochastic capacity limitations). The field remains active, connecting geometric analysis, convex optimization, and statistical decision theory and motivated by a growing range of applications.