Distributed and Decentralized Optimization Algorithms via Consensus ALADIN

TL;DR

This paper introduces Consensus ALADIN, a distributed optimization algorithm with both first- and second-order variants, capable of handling consensus constraints efficiently in centralized and decentralized settings with convergence guarantees.

Contribution

It extends the ALADIN framework to a consensus setting with a decentralized version that operates over directed graphs with quantized communication, providing convergence guarantees.

Findings

Fast local convergence with reduced communication costs.

Decentralized version handles directed graphs and quantized communication.

Convergence guarantees for convex and non-convex problems.

Abstract

Distributed optimization has found widespread applications in smart grids, optimal control, and machine learning. This paper studies distributed consensus optimization. We extend the Augmented Lagrangian-based Alternating Direction Inexact Newton (ALADIN) framework to propose Consensus ALADIN (C-ALADIN) with a central coordinator, which directly handles consensus constraints. Our C-ALADIN algorithm admits both a first-order variant and a second-order variant that employs a Hessian approximation, avoiding direct transmission of second-order information while preserving fast local convergence. We then develop a decentralized version of C-ALADIN that operates over directed graphs with quantized communication, using a finite-time coordination protocol. For both versions, we establish global convergence guarantees for convex problems and local convergence guarantees for non-convex problems. For the decentralized case, the iterates converge to a neighborhood of the optimum determined by the quantization level. Numerical results demonstrate that our methods retain fast convergence while substantially reducing communication and computational costs compared to existing decentralized approaches.