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

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.