Convergence Analysis of Alternating Direction Method of Multipliers for a Family of Nonconvex Problems
分析了交替方向乘子法（ADMM）在求解非凸共识和共享问题时的收敛性，证明当增广拉格朗日惩罚参数足够大时，经典ADMM收敛到稳定解集，且对共享问题无论变量块数多少均收敛。
The alternating direction method of multipliers (ADMM) is widely used to solve large-scale linearly constrained optimization problems, convex or nonconvex, in many engineering fields. However there is a general lack of theoretical understanding of the algorithm when the objective function is nonconvex. In this paper we analyze the convergence of the ADMM for solving certain nonconvex consensus and sharing problems. We show that the classical ADMM converges to the set of stationary solutions, provided that the penalty parameter in the augmented Lagrangian is chosen to be sufficiently large. For the sharing problems, we show that the ADMM is convergent regardless of the number of variable blocks. Our analysis does not impose any assumptions on the iterates generated by the algorithm and is broadly applicable to many ADMM variants involving proximal update rules and various flexible block selection rules.