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：考虑一种用于求解带有线性约束的非凸不可分优化问题的三块交替方向乘子法（ADMM）。受文献[1]启发，每次迭代中对第三个变量进行两次更新以确保全局收敛性。基于强库尔迪卡-沃伊托维奇性质，我们证明ADMM生成的序列全局收敛至增广拉格朗日函数的临界点。同时分别指出：当交换第一变量与第二变量的更新顺序时，以及当对第一变量添加近端项以处理更一般的不可分问题时，所提ADMM的收敛性。此外，我们对三个非凸问题进行了数值实验：多测量向量（MMV）、稳健主成分分析（RPCA）和非负矩阵补全（NMC）。实验结果表明所提ADMM的高效性与优越性。