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A Distributed Gradient-based Algorithm for Optimization Problems with Coupled Equality Constraints

Chenyang Qiu, Zongli Lin

TL;DR

This work tackles distributed optimization with coupled equality constraints by introducing a single-loop gradient-based algorithm (DGA) that eliminates the need to solve local subproblems at every iteration. Each agent performs a gradient step followed by a projection onto its local feasible set, with dual and consensus variables updated in a distributed manner using only local communications. Under convexity and Lipschitz-smoothness, the method achieves a nonergodic sublinear rate $o(1/k)$, and it attains linear convergence when the local objectives are $\mu$-strongly convex and smooth. Numerical experiments on the IEEE $118$-bus system show faster convergence and reduced computation time compared to state-of-the-art methods that require local optimization subproblems, demonstrating the approach’s practicality for scalable, privacy-preserving distributed optimization in power systems and related networks.

Abstract

This paper studies a class of distributed optimization problems with coupled equality constraints in networked systems. Many existing distributed algorithms rely on solving local subproblems via the $\operatorname{argmin}$ operator in each iteration. Such approaches become computationally burdensome or intractable when local cost functions are complex. To address this challenge, we propose a novel distributed gradient-based algorithm that avoids solving a local optimization problem at each iteration by leveraging first-order approximations and projection onto local feasible sets. The algorithm operates in a fully distributed manner, requiring only local communication without exchanging gradients or primal variables. We rigorously establish sublinear convergence for general convex cost functions and linear convergence under strong convexity and smoothness conditions. Numerical simulation on the IEEE 118-bus system demonstrates the superior computational efficiency and scalability of the proposed method compared to several state-of-the-art distributed optimization algorithms.

A Distributed Gradient-based Algorithm for Optimization Problems with Coupled Equality Constraints

TL;DR

This work tackles distributed optimization with coupled equality constraints by introducing a single-loop gradient-based algorithm (DGA) that eliminates the need to solve local subproblems at every iteration. Each agent performs a gradient step followed by a projection onto its local feasible set, with dual and consensus variables updated in a distributed manner using only local communications. Under convexity and Lipschitz-smoothness, the method achieves a nonergodic sublinear rate , and it attains linear convergence when the local objectives are -strongly convex and smooth. Numerical experiments on the IEEE -bus system show faster convergence and reduced computation time compared to state-of-the-art methods that require local optimization subproblems, demonstrating the approach’s practicality for scalable, privacy-preserving distributed optimization in power systems and related networks.

Abstract

This paper studies a class of distributed optimization problems with coupled equality constraints in networked systems. Many existing distributed algorithms rely on solving local subproblems via the operator in each iteration. Such approaches become computationally burdensome or intractable when local cost functions are complex. To address this challenge, we propose a novel distributed gradient-based algorithm that avoids solving a local optimization problem at each iteration by leveraging first-order approximations and projection onto local feasible sets. The algorithm operates in a fully distributed manner, requiring only local communication without exchanging gradients or primal variables. We rigorously establish sublinear convergence for general convex cost functions and linear convergence under strong convexity and smoothness conditions. Numerical simulation on the IEEE 118-bus system demonstrates the superior computational efficiency and scalability of the proposed method compared to several state-of-the-art distributed optimization algorithms.

Paper Structure

This paper contains 10 sections, 61 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Optimization Problem
  4. Communication Model
  5. Algorithm Development
  6. Convergence Analysis
  7. Sublinear Convergence Rate
  8. Linear Convergence Rate
  9. Numerical Experiments
  10. Conclusions

Figures (3)

  • Figure 1: The power output error of each generator under the proposed Algorithm \ref{['algorithm']}.
  • Figure 2: Comparison of optimal error over iteration for the six algorithms.
  • Figure 3: Comparison of optimal error over computation time for the six algorithms.