Distributed Optimization with Coupled Constraints over Time-Varying Digraph

Abstract: In this paper, we develop a distributed algorithm for solving a class of distributed convex optimization problems where the local objective functions can be a general nonsmooth function, and all equalities and inequalities are network-wide coupled. This type of problem arises from many areas, such as economic dispatch, network utility maximization, and demand response. Integrating the decomposition by right hand side allocation and primal-dual methods, the proposed algorithm is able to handle the distributed optimization over networks with time-varying directed graph in fully distributed fashion. This algorithm does not require the communication of sensitive information, such as primal variables, for privacy issues. Further, we show that the proposed algorithm is guaranteed to achieve an $O(1/k)$ rate of convergence in terms of optimality based on duality analysis under the condition that local objective functions are strongly convex but not necessarily differentiable, and the subdifferential of local inequalities is bounded. We simulate the proposed algorithm to demonstrate its remarkable performance.

• The paper introduces a novel primal–dual algorithm that achieves an O(1/k) convergence rate for distributed, nonsmooth convex optimization with network-coupled constraints.
• It employs a single communication step per iteration by mixing only dual variables, ensuring that local primal data remains private throughout the process.
• Numerical results demonstrate rapid primal feasibility and dual suboptimality improvements, confirming the method’s robustness across dynamically changing directed graphs.

Distributed Optimization with Coupled Constraints Over Time-Varying Digraphs

Introduction

This paper addresses a fundamental challenge in distributed convex optimization for networked multi-agent systems: optimizing a global objective function decomposed into private local objective functions, subject to a set of network-coupled equality and inequality constraints, over a time-varying directed communication topology. Such scenarios emerge in large-scale resource allocation, economic dispatch, network utility maximization, and distributed demand response, where data privacy and local autonomy must be preserved and where the network graph can be unbalanced and dynamically changing.

Consensus optimization is a well-studied paradigm; however, network-wide coupled constraints—especially in time-varying directed graphs—impose significant algorithmic complexity. Prior distributed algorithms based on dual decomposition, primal decomposition (e.g., right-hand side allocation), and primal–dual methods yield tractable formulations for undirected or static graphs, but their generalization to time-varying digraphs with strong privacy requirements is technically nontrivial.

Problem Formulation

The considered problem involves $N$ agents communicating over a time-varying directed graph $\mathcal{G}_k=(\mathcal{V},\mathcal{E}_k)$ at time $k$. Each agent $i$ controls variable $x_i\in\mathbb{R}^{d_i}$; the global objective is $\sum_i f_i(x_i)$ with strongly convex, possibly nonsmooth (but not necessarily differentiable) $f_i$, and is subject to the following network-wide coupling constraints:

• Global equality: $\sum_i A_i x_i = \sum_i b_i$
• Global inequality: $\sum_i g_i(x_i) \leq 0$ where $A_i \in \mathbb{R}^{p \times d_i}$, $\mathcal{G}_k=(\mathcal{V},\mathcal{E}_k)$0, and $\mathcal{G}_k=(\mathcal{V},\mathcal{E}_k)$1 are private information for each agent.

The primary technical challenge lies in the coupling of constraints, nonsmoothness of the local objectives, the time-varying and directed nature of the communication topology, and the stringent privacy requirement: agents are restricted from revealing their primal variables at any iteration.

Algorithmic Development

The proposed method is formulated via a right-hand side allocation strategy coupled with primal–dual augmented Lagrangian steps. By introducing auxiliary variables $\mathcal{G}_k=(\mathcal{V},\mathcal{E}_k)$2 per agent, the dense coupling constraints are transformed into separable local constraints and network-wide zero-mean consensus constraints. These are amenable to decentralized dual optimization.

A key innovation is the use of a single communication step per iteration via a time-varying doubly stochastic mixing matrix $\mathcal{G}_k=(\mathcal{V},\mathcal{E}_k)$3, uniquely suited for directed and time-varying graphs. The mixing is only applied to dual variables—ensuring that primal variables, gradients, and contributions to coupled constraints remain private throughout.

Each agent iteratively:

1. Solves a local, strongly convex subproblem augmented with Lagrangian and penalty terms, using local dual variable updates.
2. Mixes dual variables using $\mathcal{G}_k=(\mathcal{V},\mathcal{E}_k)$4.
3. Updates auxiliary variables with a fixed step-size, maintaining their adherence to network-wide zero-sum constraints via the double-stochasticity structure.

This yields a fully distributed, primal–dual algorithm generalizing primal decomposition and consensus-based dual ascent, which can operate reliably even under time-varying directed network conditions.

Convergence Analysis

The convergence analysis leverages duality theory, maximal monotone operator arguments, and aggregate lower-bounding (ALB) methodologies. Under standard convexity, Slater-type feasibility, strong connectivity, and subgradient boundedness assumptions:

• The rate of convergence in terms of dual suboptimality is shown to be $\mathcal{G}_k=(\mathcal{V},\mathcal{E}_k)$5, where $\mathcal{G}_k=(\mathcal{V},\mathcal{E}_k)$6 is the iteration index.
• Primal feasibility and global optimality for the original variable allocations are established as a corollary of the dual convergence and the strong convexity structure.
• The proof accommodates arbitrary, bounded-norm, possibly large time variations in the digraph sequence, provided doubly stochastic mixing is preserved.

Crucially, this result is obtained without requiring diminishing step sizes or additional rounds of communication.

The $\mathcal{G}_k=(\mathcal{V},\mathcal{E}_k)$7 convergence rate is significant for distributed non-smooth strongly convex optimization with coupled constraints, and matches or exceeds the best existing guarantees under weaker privacy and network assumptions.

Numerical Results

The paper provides simulations with random strongly convex objectives (including $\mathcal{G}_k=(\mathcal{V},\mathcal{E}_k)$8-regularization) and high-dimensional coupled constraints for $\mathcal{G}_k=(\mathcal{V},\mathcal{E}_k)$9 agents. The communication graph is dynamically evolving and directed. Empirical results demonstrate:

• Rapid decay of primal feasibility gap and dual suboptimality.
• Robustness with respect to strong variations in the network topology.
• Preservation of privacy, as no primal information is exchanged.

These results confirm the algorithmic claims and underpin its potential practical deployment in privacy-critical, large-scale cyber-physical and multi-robot systems.

Theoretical and Practical Implications

The approach synthesizes techniques from primal decomposition, dual consensus, and operator theory to achieve strong optimality guarantees in arguably the most challenging distributed optimization setting: nonsmooth, constraint-coupled problems over time-varying digraphs with privacy constraints. The algorithm requires neither primal nor gradient communication, marking it as suitable for privacy-critical domains.

The theoretical development elucidates the interplay between consensus tracking in the dual space, augmented Lagrangian regularization, and the role of double stochasticity in maintaining zero-sum auxiliary constraints.

Future research directions include weakening the double stochasticity requirement (possibly replacing with push-sum or weighted-average methods), extending to asynchronous updates, and handling stochastic or adversarial communication failures.

Conclusion

The paper presents a distributed primal–dual optimization algorithm for coupled, constrained, nonsmooth convex programs under time-varying directed communication. By leveraging a careful decomposition, augmented Lagrangian regularization, and consensus in the dual space, it achieves an $k$0 convergence rate in dual optimality with strong privacy guarantees. The framework advances the frontier of distributed constrained convex optimization, providing a rigorous foundation and practical methodology for complex networked systems where communication, topology, and privacy constraints are all critical (2604.10422).