Coupled Equality Constraints in Networks

• Coupled equality constraints in networked systems are global affine relationships linking agents’ decisions to ensure feasibility in decentralized optimization.
• Modern methodologies apply convex duality, consensus reformulations, and quadratic penalties to achieve accelerated convergence and minimal communication overhead.
• Violation-free techniques, including slack-variable reformulations and nonlinear mappings, guarantee exact feasibility and are well-suited for safety-critical applications.

Coupled equality constraints in networked systems refer to global affine constraints that simultaneously link the decisions of multiple subsystems or agents in a distributed optimization context. These constraints—typically of the form $\sum_{i=1}^N A_i x_i = b$, with $x_i$ local decision vectors and $A_i$ agent-specific matrices—manifest in applications like power system dispatch, cooperative control, and network flow. The handling of such constraints in a distributed setup is critical for ensuring feasibility and optimality without centralized coordination, especially when local information and computation are preserved. Recent advancements have yielded accelerated, communication-efficient, and violation-free algorithms with provable rates, leveraging convex duality, consensus protocols, augmented Lagrangian techniques, and relaxation/duality reformulations.

1. Formal Problem Definition and Motivations

In distributed convex optimization over a network of $n$ agents, coupled equality constraints arise when local variables $(x_i)$ are linked via global affine relations, e.g.,

$$\min_{x \in X} \;\sum_{i=1}^n f_i(x_i) \quad \text{s.t.} \quad \sum_{i=1}^n A_i x_i = b,$$

with $x_i \in \mathbb{R}^{p_i}$, $f_i$ convex, and $X = X_1 \times \cdots \times X_n$ compact. The coupling often extends to nonseparable objectives or additional global nonlinear inequalities, necessitating algorithms that can enforce both optimality and strict constraint satisfaction in a decentralized manner (Qiu et al., 24 Nov 2025, Qiu et al., 24 Nov 2025, Liu et al., 11 Apr 2024).

Motivating contexts include:

• Economic dispatch: enforcing global power balance among distributed generators.
• Resource allocation: respecting total supply-demand constraints in distributed logistics.
• Distributed model predictive control (DMPC): enforcing networked state and input constraints for multi-agent dynamical systems.

2. Duality, Consensus Reformulations, and Quadratic Penalties

To enable distributed enforcement, a duality-based approach is employed. By introducing multipliers $\mu$ (for affine constraints) and possibly $\delta$ (for inequality constraints), the partially dualized problem becomes

$$L(x, \mu, \delta) = \sum_i f_i(x_i) + \langle \mu, A_i x_i - b_i \rangle + \cdots$$

with the dual subproblem separable across nodes. Achieving consensus among the dual variables across agents is cast as an optimization with a quadratic Laplacian penalty on the disagreement,

$\frac{\theta}{2}\|W^{1/2}y\|^2,$

where $W = L \otimes I$ encodes the network topology (Qiu et al., 24 Nov 2025). Algorithms such as Nesterov-accelerated linearized methods of multipliers or distributed saddle-point procedures exploit this structure, updating local dual variables, enforcing consensus via weighted neighbor exchanges, and iteratively reducing both optimality gaps and constraint violations.

Table: Core distributed dual consensus strategy | Step | Description | Source | |------------------------------|-------------------------------------------|--------------------| | Local primal solve | Oracle/minimization under dual gradient | (Qiu et al., 24 Nov 2025) | | Dual variable consensus | Neighborhood averaging / Laplacian penalty| (Qiu et al., 24 Nov 2025, Qiu et al., 24 Nov 2025) | | Multiplier update | Proximal/gradient step on dual residual | (Qiu et al., 24 Nov 2025) |

Explicit rates depend on the network's spectral gap ($\lambda_2(W)$) and the smoothness and strong convexity constants of $f_i$. Quadratic penalties on consensus violation guarantee asymptotic agreement and, by strong duality, drive primal residuals to zero at an explicit, typically $O(1/N^2) + O(1/N)$ non-ergodic rate (Qiu et al., 24 Nov 2025).

3. Distributed First-Order, Proximal, and Gradient-Based Algorithms

Recent approaches avoid costly inner minimizations by employing local linearizations and explicit projections:

• Distributed gradient-based algorithms linearize $f_i$ at current iterates, combine with dual-tracking terms, and project onto the feasible local constraint set (Qiu et al., 24 Nov 2025).
• Each iteration involves local gradient steps, followed by consensus-based updates to dual variables, and communication of only lightweight state (e.g., dual multipliers).
• For box or simple affine constraints, explicit projection formulas enable scalable closed-form updates.

Under L-smoothness and convexity, sublinear $O(1/k)$ convergence rates are obtained; for strongly convex and smooth objectives, geometric (linear) rates are proved. The algorithms scale efficiently on large systems (e.g., IEEE-118 bus) and outperform state-of-the-art AL and ADMM-based methods both in iteration count and per-iteration complexity (Qiu et al., 24 Nov 2025).

4. Exact Constraint Satisfaction and Violation-Free Algorithms

Conventional primal–dual or dual consensus schemes may attain exact feasibility only asymptotically. For safety-critical or system-theoretic contexts, algorithms with recursive, all-time constraint satisfaction are indispensable:

• Slack-variable reformulations introduce local auxiliary variables $y^{[m]}$ to localize the global constraint, together with network-dependent mappings $(I - P^{[m]})y^{[m]}$ (where $P^{[m]}$ is a doubly-stochastic matrix on the induced constraint subgraph). It follows that, at every iteration, the primal variables $x(t)$ are feasible (i.e., $\sum_i A_i x_i(t) = b$) (Liu et al., 11 Apr 2024).
• The distributed algorithm alternates between local solves augmented by these slacks, neighbor exchange of dual variables, and gradient (or accelerated dual averaging) updates. Convergence to global optimum is guaranteed with $O(1/t^2)$ (strongly convex) or $O(1/\sqrt{t})$ (merely convex) rates.

The zero-violation property underpins applications such as distributed control barrier function (CBF)-forwarded safety-critical consensus, enabling distributed CBF-QP solves that mirror centralized feasibility at all times (Liu et al., 11 Apr 2024).

5. Extensions: Delays, Compression, Variable Coupling, and Nonlinearities

Robustness to communication imperfections and richer coupling topologies has been addressed:

• Algorithms incorporating spatio-temporally compressed communication, using compressive/quantization mappings on exchanged states, retain linear convergence under strong convexity provided the compressors are sector-bounded and errors are filtered via stable ODEs (Ren et al., 4 Mar 2025).
• Variable coupling, where local objectives and constraints depend on both self and neighboring variables, is handled by a projected primal-dual framework leveraging virtual queues and augmented-Lagrangian-inspired local gradients. This approach achieves $O(1/k)$ rates for both coupled equality and inequality constraints in a fully decentralized way with only neighbor-to-neighbor messaging (Wang et al., 15 Jul 2024).
• Nonlinearities, delays, and switching topologies are addressed through nonlinear Laplacian-gradient flow protocols. Sum-preserving nonlinear mappings $g(\cdot)$ enable robust, anytime-feasible adjustment of resource constraints under quantization, saturation, or impulsive disturbance, as well as in the presence of heterogeneous, bounded, time-varying communication delays (Doostmohammadian et al., 2023).

6. Alternative Approaches: Relaxation, PDMM, and Singular Perturbation

Alternative frameworks include:

• Relaxation–duality schemes add explicit, penalized slack variables to the coupling constraint, transforming constrained dual problems into compact, consensus-coupled local duals with efficient local solves and sublinear convergence, while ensuring all iterates converge to the primal optimal solution without averaging (Notarnicola et al., 2017).
• Primal–dual method of multipliers (PDMM) with Peaceman–Rachford splitting refines network-wide dual updates for coupled constraints, supporting both equality and inequality couplings, and handling asynchronous communication and losses (Heusdens et al., 2023).
• Singular perturbation approaches decompose the networked optimization into fast dynamic consensus subsystems (tracking global residuals) and slow primal–dual update dynamics. This enables exact enforcement of global constraints, privacy preservation, and robustness, with convergence assured under strong convexity, Slater conditions, and graph connectivity (Hoang et al., 2017).

7. Practical Performance, Theoretical Guarantees, and Applications

Empirical results across various testbeds (e.g., IEEE-118 bus, distributed model predictive control, resource allocation) demonstrate that modern algorithms for coupled equality constraints achieve:

• Superior convergence speed relative to contemporary AL, ADMM, and gradient-tracking baselines, with rapid feasibility decay and suboptimality reduction (Qiu et al., 24 Nov 2025, Qiu et al., 24 Nov 2025, Qiu et al., 24 Nov 2025).
• Strict enforcement of global constraints for all iterates in safety-critical contexts when using slack-based or nonlinear sum-preserving approaches (Liu et al., 11 Apr 2024, Doostmohammadian et al., 2023).
• High scalability and computational efficiency due to avoidance of nested or centralized optimization, with per-iteration complexity dominated by local gradient/proximal steps and lightweight communication.
• Extension to heterogeneous objectives, variable coupling, constrained resource scheduling, and distributed CBF-based safety control.

An explicit summary is provided in the following table:

Property	Realization/Guarantee	Reference
Non-ergodic O($1/N^2$)+O($1/N$) rate	Accelerated Nesterov-type dual method	(Qiu et al., 24 Nov 2025)
Sublinear/linear convergence	Gradient-based projection, strongly convex	(Qiu et al., 24 Nov 2025)
All-time feasibility	Slack-variable, violation-free approach	(Liu et al., 11 Apr 2024)
Robustness to compression/delay	Spatio-temporal compression, nonlinear flows	(Ren et al., 4 Mar 2025, Doostmohammadian et al., 2023)
Applicability	Power, resource, DMPC, CBFs, variable coupling	(Qiu et al., 24 Nov 2025, Wiltz et al., 2022, Wang et al., 15 Jul 2024)

These developments establish a rigorous foundation and design principles for distributed optimization in networked systems subject to coupled equality constraints, with ongoing research extending these tools to broader nonconvex, stochastic, or adversarially-perturbed scenarios.