Convergence of Generalized Alternating Direction Method of Multipliers for Nonseparable Nonconvex Objective with Linear Constraints

Abstract:

      In this paper, we consider the convergence of the generalized alternating direction method of multipliers (GADMM) for solving linearly constrained nonconvex minimization model whose objective contains coupled functions. Under the assumption that the augmented Lagrangian function satisfies the Kurdyka-{\L}ojasiewicz inequality, we prove that the sequence generated by the GADMM converges to a critical point of the augmented Lagrangian function when the penalty parameter in the augmented Lagrangian function is sufficiently large. Moreover, we also present some sufficient conditions guaranteeing the sublinear and linear rate of convergence of the algorithm.