# Tracking-ADMM for Distributed Constraint-Coupled Optimization

## TL;DR

This paper introduces a distributed optimization algorithm combining dynamic average consensus with ADMM to solve constraint-coupled problems, ensuring convergence to optimal solutions in a networked setting.

## Contribution

It develops a novel fully distributed scheme that tracks constraint violations dynamically within ADMM, enhancing distributed optimization methods.

## Key findings

- Proves convergence of primal and dual estimates to optimal solutions.
- Demonstrates the effectiveness through a numerical case study.
- Ensures consensus and asymptotic optimality in a distributed setting.

## Abstract

We consider constraint-coupled optimization problems in which agents of a network aim to cooperatively minimize the sum of local objective functions subject to individual constraints and a common linear coupling constraint. We propose a novel optimization algorithm that embeds a dynamic average consensus protocol in the parallel Alternating Direction Method of Multipliers (ADMM) to design a fully distributed scheme for the considered set-up. The dynamic average mechanism allows agents to track the time-varying coupling constraint violation (at the current solution estimates). The tracked version of the constraint violation is then used to update local dual variables in a consensus-based scheme mimicking a parallel ADMM step. Under convexity, we prove that all limit points of the agents' primal solution estimates form an optimal solution of the constraint-coupled (primal) problem. The result is proved by means of a Lyapunov-based analysis simultaneously showing consensus of the dual estimates to a dual optimal solution, convergence of the tracking scheme and asymptotic optimality of primal iterates. A numerical study on optimal charging schedule of plug-in electric vehicles corroborates the theoretical results.