This paper proposes a proximal variant of the alternating direction method of multipliers (ADMM) for distributed optimization. Although the current versions of ADMM algorithm provide promising numerical results in producing solutions that are close to optimal for many convex and non-convex optimization problems, it remains unclear if they can converge to a stationary point for weakly convex and locally non-smooth functions. Through our analysis using the Moreau envelope function, we demonstrate that MADM can indeed converge to a stationary point under mild conditions. Our analysis also includes computing the bounds on the amount of change in the dual variable update step by relating the gradient of the Moreau envelope function to the proximal function. Furthermore, the results of our numerical experiments indicate that our method is faster and more robust than widely-used approaches.

翻译：本文提出了一种用于分布式优化的交替方向乘子法（ADMM）的近端变体。尽管当前版本的ADMM算法在许多凸和非凸优化问题中能提供接近最优解的良好数值结果，但对于弱凸且局部非光滑函数，其是否能收敛到驻点仍不明确。通过使用Moreau包络函数的分析，我们证明MADDM在温和条件下确实可以收敛到驻点。我们的分析还包括通过将Moreau包络函数的梯度与近端函数相关联，计算对偶变量更新步中变化量的界。此外，数值实验结果表明，我们的方法比广泛使用的现有方法更快且更稳健。