# Complexity Certification of a Distributed Augmented Lagrangian Method

## TL;DR

This paper provides complexity bounds for a distributed Augmented Lagrangian algorithm solving convex problems with coupled constraints, demonstrating an $O(1/\epsilon)$ iteration complexity for achieving near-optimal solutions.

## Contribution

It introduces the ADAL algorithm with explicit complexity bounds and a method to select stepsizes, advancing distributed convex optimization theory.

## Key findings

- ADAL achieves $O(1/\epsilon)$ complexity for $\ ext{epsilon}$-optimal solutions.
- Provides an explicit upper bound for the dual multiplier in distributed settings.
- Demonstrates applicability to model predictive control problems with networked subsystems.

## Abstract

In this paper we present complexity certification results for a distributed Augmented Lagrangian (AL) algorithm used to solve convex optimization problems involving globally coupled linear constraints. Our method relies on the Accelerated Distributed Augmented Lagrangian (ADAL) algorithm, which can handle the coupled linear constraints in a distributed manner based on local estimates of the AL. We show that the theoretical complexity of ADAL to reach an $\epsilon$-optimal solution both in terms of suboptimality and infeasibility is $O(\frac{1}{\epsilon})$ iterations. Moreover, we provide a valid upper bound for the optimal dual multiplier which enables us to explicitly specify these complexity bounds. We also show how to choose the stepsize parameter to minimize the bounds on the convergence rates. Finally, we discuss a motivating example, a model predictive control (MPC) problem, involving a finite number of subsystems which interact with each other via a general network.