A partial Bregman ADMM with a general relaxation factor for structured nonconvex and nonsmooth optimization

Abstract

In this paper, a partial Bregman alternating direction method of multipliers (ADMM) with a general relaxation factor \(\alpha \in (0,\frac{1+\sqrt{5}}{2})\) is proposed for structured nonconvex and nonsmooth optimization, where the objective function is the sum of a nonsmooth convex function and a smooth nonconvex function without coupled variables. We add a Bregman distance to alleviate the difficulty of solving the nonsmooth subproblem. For the smooth subproblem, we directly perform a gradient descent step of the augmented Lagrangian function, which makes the computational cost of each iteration of our method very cheap. To our knowledge, the nonconvex ADMM with a relaxation factor \(\alpha \ne 1\) in the literature has never been studied for the problem under consideration. Under some mild conditions, the boundedness of the generated sequence, the global convergence and the iteration complexity are established. The numerical results verify the efficiency and robustness of the proposed method.