Capacity-Constrained Feasible Set

• Capacity-Constrained Feasible Set is defined as the intersection of an unconstrained decision space with explicit capacity limits, ensuring adherence to physical and resource constraints.
• It is applied across diverse domains such as information theory, optimal transport, power systems, and control, where system performance is optimized under hard resource bounds.
• Its detailed geometric, duality, and algorithmic structures provide actionable insights for designing robust and computationally efficient decision-making frameworks.

A capacity-constrained feasible set refers to the set of decision configurations, trajectories, measures, or controls that obey both the structural or physical constraints of a system and explicit capacity (upper or lower) limits on resources, flows, or actions. This concept is fundamental across information theory, optimal transport, power systems, network flows, optimization, control theory, and economics, capturing the interplay between physical realizability and hard resource bounds. Below, key theoretical and applied frameworks are surveyed.

1. Formal Definitions and Structures

The capacity-constrained feasible set is always an intersection of the unconstrained feasible set (determined by governing equations or topology) and a set of inequalities encoding capacity limits.

• Information theory (input-constrained channels):

For an irreducible finite-type constraint $S$ defined by forbidden patterns $F \subset \mathcal{X}^{m+1}$ ($\mathcal{X} = \{0,1\}$), the input feasible set consists of all infinite sequences $x \in \mathcal{X}^{\mathbb{Z}}$ so that every finite subblock belongs to $S$—i.e., no forbidden pattern occurs (0803.3360).

• Optimal transport (continuous or discrete):

With marginals $\mu \in P(X)$, $\nu \in P(Y)$ and a measurable capacity density $\bar h: X \times Y \to [0, \infty)$, the feasible set is

$$\mathcal{H} = \left\{ \gamma \in \mathcal{P}(X \times Y): \pi^X_\#\gamma = \mu,~ \pi^Y_\#\gamma = \nu,~ 0 \leq \frac{d\gamma}{dx\,dy} \leq \bar h(x, y) \text{ a.e.} \right\}$$

(Korman et al., 2013, Wu et al., 2022).

• Power systems (AC or DC networks):

The feasible set is the set of power injections or flows $(p, q)$ such that there exists a system state (voltages, currents) satisfying both the nonlinear network equations and operational (capacity) bounds on voltages, currents, and line flows (Dvijotham et al., 2015, Zhang et al., 2023, Nazir et al., 2019).

• Facility location:

For $n$ agents, $m$ facilities (capacities $c_j$), and assignment variables $s_{i, j}$,

$$\mathcal{F} = \left\{(y, s): \sum_{j=1}^m s_{i, j} = 1 ~\forall i;~ \sum_{i=1}^n s_{i, j} \leq c_j ~\forall j;~ s_{i, j} \in \{0,1\}\right\}$$

(Walsh, 2020).

• Control systems with actuation bounds:

For a finite-horizon linear system $\dot x = A(t)x + B(t)u$, initial/terminal state constraints, and $|u(t)| \leq M$, the feasible set is $\mathcal{F}_M = \mathcal{X} \cap \mathcal{U}_M$ where $\mathcal{X}$ is the set of steering controls and $\mathcal{U}_M$ is the $L^\infty$-ball in control space (Burachik et al., 2024).

Capacity-constrained feasible sets are generally convex polytopes (when constraints and structure are linear) or more complicated convex/possibly disconnected sets (for networks, nonlinear constraints, or integer assignments).

2. Geometry and Extreme Points

The geometric and algebraic structure of the capacity-constrained feasible set is essential for characterizing solutions.

• Polytopality and extreme points:

In discrete settings (optimal transport, facility location, network flow), the capacity-constrained set is a bounded convex polytope. For capacity-constrained optimal transport, extreme points are characterized by the “all-or-nothing” (bang–bang) principle: at almost every location, the optimal plan is either zero or saturates the local capacity bound, i.e., $h(x, y) \in \{0, \bar h(x, y)\}$ almost everywhere (Korman et al., 2013). In discrete transport, extreme points correspond to assignments where the box constraints $0 \leq \pi_{ij} \leq U_{ij}$ are tight at as many entries as possible (Wu et al., 2022).

• Piecewise structure and combinatorics:

In capacity-constrained facility location, the feasible set is typically a union of exponentially many disconnected polyhedral cells (one per capacity-respecting agent-to-facility assignment), often highly non-convex, with special one-dimensional simplification arising from interval assignments (Walsh, 2020).

• Network and control:

In power flow applications, the feasible injection region under capacity constraints typically becomes nonconvex or disconnected for networks, motivating convex inner or outer approximations for operational tractability (Dvijotham et al., 2015, Zhang et al., 2023, Nazir et al., 2019).

3. Duality, Regularization, and Uniqueness

Capacity constraints enter dual formulations directly and have fundamental consequences for uniqueness and solution selection.

• Optimal transport duality:

The dual of the linear program for continuous or discrete capacity-constrained optimal transport introduces Lagrange multipliers for the mass and box constraints. In the regularized discrete setting, double-entropy regularization automatically keeps solutions in the interior of the capacity box—no explicit projections are needed (Wu et al., 2022).

• Bang–bang uniqueness:

When the objective is linear and the feasible set is convex, extremality implies uniqueness: under capacity constraints, only the extreme points (bang–bang solutions) can be optimal, and construction of marginal-preserving perturbations precludes optimality for interior points (Korman et al., 2013).

• Information-theoretic channels:

Capacity-constrained feasible sets, such as those governed by input constraints, admit maximization over Markov processes that realize the topological entropy of the allowed set. The Parry measure, maximizing entropy rate subject to the constraint graph, achieves the channel capacity (0803.3360).

• Legendre duals in information theory:

The cost-constrained channel capacity (e.g., Augustin capacity) is the supremum of information rate over cost-bounded input distributions. Dualizing in the cost constraint yields an unconstrained maximization over a Lagrangian with a “cost-weight” parameter, and the set of optimal measures (centers) varies smoothly with the budget in the interior of the feasible set (Nakiboglu, 2018).

4. Asymptotics, Structural Properties, and Algorithms

Asymptotic analysis and constructive algorithms enable explicit characterization and approximation of capacity-constrained feasible sets.

• Asymptotic expansion in input-constrained BSCs:

For a BSC($\varepsilon$) under a finite-type input constraint $S$,

$$C(S, \varepsilon) = \log\lambda - K \varepsilon\log(1/\varepsilon) + O(\varepsilon)$$

where $\lambda$ is the Perron–Frobenius eigenvalue of the constraint graph, $K=1-f(X_{p_{\max}})$, and $f(\cdot)$ quantifies the probability that a single bit-flip creates a forbidden word (0803.3360).

• Convex approximations:

In power systems, semidefinite and second-order cone relaxations provide convex inner approximations (typically polytopes or ellipsoids) of capacity-constrained feasible sets. These approximations are tight in the sense that any further uniform expansion leads to infeasibility in the original nonlinear system (Dvijotham et al., 2015, Zhang et al., 2023, Nazir et al., 2019).

• Regularized algorithms:

Double entropic regularization formulations in capacity-constrained optimal transport lead to efficient algorithms where fast root-finding or Newton iteration is used for enforcing capacity and marginal constraints, with guaranteed convergence to a strictly feasible solution (Wu et al., 2022).

• Iterative methods:

Iterative refinement is crucial for tightening convex inner approximations to the true feasible set in AC power flow problems—using updated base points to maximize the inclusion within the nonconvex feasible locus (Zhang et al., 2023).

5. Impact on System Behavior, Performance, and Structure

Imposing capacity constraints induces qualitative and quantitative changes in system performance.

• Reduced deliverability:

In BSCs or muscle capacity models, capacity constraints reduce achievable rates or output forces—often directionally, reflecting specific vulnerabilities or safety margins under the restrictions (0803.3360, Rezzoug et al., 2023).

• Structural sparsity or localization:

In optimal transport, extremal plans under capacity constraints display spatially localized “all-or-nothing” structure, resulting in sharp thresholds, islands, and disconnected phenomena in the feasible set support (Korman et al., 2013).

• Changed coordination and resource allocation:

In multi-agent, economic, or engineering systems, constraints fundamentally change optimal assignments and coordination, often concentrating flows or allocations and excluding globally optimal unconstrained configurations (Dubois et al., 2021, Clark, 2024).

• Criticality and infeasibility margins:

In control, the critical minimal bound $M_{\mathrm{crit}}$ for system controllability provides a strict threshold: below this, feasibility is lost and the best-approximation problem becomes nontrivial, often selecting discontinuous (bang–bang) or switching-type controls (Burachik et al., 2024).

6. Applications and Case Studies

Capacity-constrained feasible sets are foundational in a wide array of domains:

Domain	Capacity-Constrained Set	Key Result or Formula
Noisy channel coding	Inputs constrained to $S$	$$C(S,\varepsilon)=\log\lambda-K\varepsilon\log(1/\varepsilon)+O(\varepsilon)$$
Optimal transport	Joint measure $\leq \bar h(x,y)$	Extreme points: $h(x,y)\in\{0, \bar h(x,y)\}$
Power network security	Power injections, op. constraints	Inner approximations via SDP/SOCP, convex DSF $S$
Facility location	Assignments, capacity on facilities	Disconnected union of assignments; interval/contiguous block assignments
TCL (demand response)	Aggregate profiles defined by bound	Polyhedral feasible set: $A\psi=b,~C\psi\leq d$
Mechanism design	Assignment, capacity, strategyproof	Pareto/frontier reduced by constraint; explicit (often unique) extreme points
Control (actuation)	$L^\infty$-ball intersection	Critical $M_{\mathrm{crit}}$; structure of gap, bang–bang controls

In each application, specific analytic and computational techniques are tailored to the underlying problem structure and nature of the capacity constraints.

7. Open Problems and Theoretical Generalizations

The study of capacity-constrained feasible sets continues to generate important questions and generalizations.

• Beyond linear and sum-type constraints: Most explicit methods address linear, sum, or box-type capacity constraints—extension to Boolean, logical, or nonlinear capacity rules is active (Dubois et al., 2021).
• Nonconvexity and tractable relaxation: In nonlinear network or structural dynamics, feasible sets are often nonconvex or disconnected. Efficient inner (or outer) convex relaxations remain central, both for decision support and robust operations (Zhang et al., 2023, Dvijotham et al., 2015).
• Scaling laws and sensitivity: In principal–agent models, scaling laws ($\alpha(C)$ reduction in reward/output) under capacity constraints provide sharp welfare and performance sensitivity; similar scaling principles may have analogs in other domains (Clark, 2024).
• Robustness to uncertainty: For power systems and aggregation, maintaining feasibility under stochastic or uncertain parameters while respecting capacity constraints is inherently linked to the geometry and measure of the feasible set (Dvijotham et al., 2015, Coffman et al., 2019).

The analysis and computation of capacity-constrained feasible sets are central to the rigorous design, optimization, and control of large-scale engineered and economic systems. The domain continues to evolve toward ever-more tractable and robust characterization under increasing complexity of constraints and uncertainty.