A Symmetric Bregman Alternating Direction Method of Multipliers for Separable Nonconvex Minimization Problem

The alternating direction method of multipliers (ADMM) and its symmetric version are efficient for minimizing two-block separable problems with linear constraints. However, both ADMM and symmetric ADMM have limited versatility across various fields due to the requirement that the gradients of differentiable functions exhibit global Lipschitz continuity, a condition that is typically challenging to satisfy in nonconvex optimization problems. Recently, a novel Bregman ADMM that not only eliminates the need for global Lipschitz continuity of the gradient, but also ensures that Bregman ADMM can be degenerated to the classical ADMM has been proposed for two-block nonconvex optimization problems with linear constraints. Building on this, we propose a symmetric Bregman alternating direction method of multipliers, which can be degenerated into the symmetric ADMM and the Bregman ADMM, and thus further degenerated into the classical ADMM. Moreover, when solving separable nonconvex optimization problems, it does not require the global Lipschitz continuity of the gradients of differentiable functions. Furthermore, we demonstrate that under the Kurdyka-Lojasiewicz inequality and certain conditions, the iterative sequence generated by our algorithm converges to a critical point of the problem. In addition, we examine the convergence rate of the algorithm.