D-ripALM: A Tuning-friendly Decentralized Relative-Type Inexact Proximal Augmented Lagrangian Method

TL;DR

D-ripALM is a decentralized optimization method that uses a relative error criterion for improved robustness and tuning, effectively handling both smooth and nonsmooth convex problems in multi-agent networks.

Contribution

It introduces a novel double-loop decentralized algorithm with a relative error criterion, enhancing robustness and tuning flexibility without requiring smoothness or strong convexity.

Findings

Demonstrates improved numerical robustness and tuning flexibility.

Achieves high-precision solutions with fewer communication rounds.

Provides rigorous convergence guarantees under general convexity.

Abstract

This paper proposes D-ripALM, a Decentralized relative-type inexact proximal Augmented Lagrangian Method for consensus convex optimization over multi-agent networks. D-ripALM adopts a double-loop distributed optimization framework that accommodates a wide range of inner solvers, enabling efficient treatment of both smooth and nonsmooth objectives. In contrast to existing double-loop distributed augmented Lagrangian methods, D-ripALM employs a relative-type error criterion to regulate the switching between inner and outer iterations, resulting in a more practical and tuning-friendly algorithmic framework with enhanced numerical robustness. Moreover, we establish rigorous convergence guarantees for D-ripALM under general convexity assumptions, without requiring smoothness or strong convexity conditions commonly imposed in the distributed optimization literature. Numerical experiments further demonstrate the tuning-friendly nature of D-ripALM and its efficiency in attaining high-precision solutions with fewer communication rounds.