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In this paper, we formally analyze global convergence in the realm ofdistributed consensus optimization. Current solutions have explored suchanalysis, particularly focusing on consensus alternating direction method ofmultipliers (CADMM), including convex and non-convex cases. While such effortson non-convexity offer elegant theory guaranteeing global convergence, theyentail strong assumptions and complicated proof techniques that areincreasingly pose challenges when adopted to real-world applications. Toresolve such tension, we propose a novel bi-level globalization strategy thatnot only guarantees global convergence but also provides succinct proofs, allwhile requiring mild assumptions. We begin by adopting such a strategy toperform global convergence analysis for the non-convex cases in C-ADMM. Then,we employ our proposed strategy in consensus augmented Lagrangian basedalternating direction inexact Newton method (C-ALADIN), a more recent andgeneralization of C-ADMM. Surprisingly, our analysis shows that C-ALADINglobally converges to local optimizer, complementary to the prior work onC-ALADIN, which had primarily focused on analyzing local convergence fornon-convex cases.

A Bi-level Globalization Strategy for Non-convex Consensus ADMM and ALADIN