# Asynchronous Distributed Method of Multipliers for Constrained Nonconvex Optimization

## TL;DR

This paper introduces an asynchronous distributed optimization algorithm for nonconvex constrained problems over networks, leveraging the Method of Multipliers with uncoordinated node updates and a distributed logic-AND for convergence.

## Contribution

It proposes a novel asynchronous distributed algorithm based on the Method of Multipliers, suitable for nonconvex problems, with proven convergence properties.

## Key findings

- Algorithm converges under nonconvex constraints.
- Nodes perform local descent or ascent steps asynchronously.
- Distributed logic-AND effectively coordinates node updates.

## Abstract

This paper addresses a class of constrained optimization problems over networks in which local cost functions and constraints can be nonconvex. We propose an asynchronous distributed optimization algorithm, relying on the centralized Method of Multipliers, in which each node wakes up in an uncoordinated fashion and performs either a descent step on a local Augmented Lagrangian or an ascent step on the local multiplier vector. These two phases are regulated by a distributed logic-AND, which allows nodes to understand when the descent on the (whole) Augmented Lagrangian is sufficiently small. We show that this distributed algorithm is equivalent to a block coordinate descent algorithm for the minimization of the Augmented Lagrangian followed by an update of the whole multiplier vector. Thus, the proposed algorithm inherits the convergence properties of the Method of Multipliers.