CADMM-Prox: A Bi-level Consensus ADMM for Non-smooth Non-convex Distributed Consensus Optimization

Non-smooth and non-convex optimization problems are pervasive in machine learning, control, and signal processing, due to the need for sparse solutions and the inherently non-convex nature of many objective functions. In this paper, we study non-smooth and non-convex distributed optimization problems. We propose a novel bi-level Consensus Alternating Direction Method of Multipliers (ADMM) algorithm, termed CADMM-Prox. The proposed algorithm integrates classical Consensus ADMM with a proximal mechanism by introducing a sufficiently large proximal term associated with an outer-level variable. Under the mild assumption that the local objective functions are semi-convex, CADMM-Prox is guaranteed to converge globally to a neighborhood of a Clarke stationary point. Numerical experiments on a phase retrieval problem demonstrate that our proposed method exhibits more stable convergence behavior compared with baseline algorithm.