# Cluster-based Distributed Augmented Lagrangian Algorithm for a Class of Constrained Convex Optimization Problems

## TL;DR

This paper introduces a novel distributed continuous-time algorithm for constrained convex optimization over clustered networks, significantly reducing communication and computation costs while ensuring convergence under various convexity conditions.

## Contribution

It presents a new cluster-based distributed augmented Lagrangian algorithm with proven convergence properties and explicit bounds for inequality constraint handling.

## Key findings

- Converges asymptotically for convex costs.
- Achieves exponential convergence for strongly convex costs.
- Demonstrated effectiveness through a numerical example.

## Abstract

We propose a distributed solution for a constrained convex optimization problem over a network of clustered agents each consisted of a set of subagents. The communication range of the clustered agents is such that they can form a connected undirected graph topology. The total cost in this optimization problem is the sum of the local convex costs of the subagents of each cluster. We seek a minimizer of this cost subject to a set of affine equality constraints, and a set of affine inequality constraints specifying the bounds on the decision variables if such bounds exist. We design our distributed algorithm in a cluster-based framework which results in a significant reduction in communication and computation costs. Our proposed distributed solution is a novel continuous-time algorithm that is linked to the augmented Lagrangian approach. It converges asymptotically when the local cost functions are convex and exponentially when they are strongly convex and have Lipschitz gradients. Moreover, we use an $\epsilon$-exact penalty function to address the inequality constraints and derive an explicit lower bound on the penalty function weight to guarantee convergence to $\epsilon$-neighborhood of the global minimum value of the cost. A numerical example demonstrates our results.