We consider a three-block alternating direction method of multipliers (ADMM) for solving the nonconvex nonseparable optimization problem with linear constraint. Inspired by [1], the third variable is updated twice in each iteration to ensure the global convergence. Based on the powerful Kurdyka-Lojasiewicz property, we prove that the sequence generated by the ADMM converges globally to the critical point of the augmented Lagrangian function. We also point out the convergence of proposed ADMM with swapping the update order of the first and second variables, and with adding a proximal term to the first variable for more general nonseparable problems, respectively. Moreover, we make numerical experiments on three nonconvex problems: multiple measurement vector (MMV), robust PCA (RPCA) and nonnegative matrix completion (NMC). The results show the efficiency and outperformance of proposed ADMM.

翻译：我们考虑一种三块交替方向乘子法（ADMM），用于求解带有线性约束的非凸非可分优化问题。受文献[1]启发，每个迭代中第三变量被更新两次以确保全局收敛。基于强大的Kurdyka-Lojasiewicz性质，我们证明了该ADMM生成的序列全局收敛至增广拉格朗日函数的临界点。我们还分别指出所提ADMM在交换第一和第二变量更新顺序、以及为更一般的非可分问题对第一变量添加邻近项时的收敛性。此外，我们对三个非凸问题进行了数值实验：多测量矢量（MMV）、稳健PCA（RPCA）和非负矩阵补全（NMC）。结果表明所提ADMM的效率与优越性。