A Majorized Bregman ADMM for 3-block Linearly Constrained Difference-of-convex Problems

(Communicated by Jie Sun) In this paper, we consider the 3-block linearly constrained difference-of-convex (DC) optimization problems:

$\min_{x,y,z}\ \{ \ f(x) + g (y) + h(z) \mid A x +By+Cz =b \ \},$ (1) where

$f(x) = f_1(x)-f_2(x)~ ~\text{and}~~ g(y) = g_1(y)-g_2(y),$

$h:\mathbb{R}^{n_3}\rightarrow \mathbb{R}$ be a continuous differentiable convex function with Lipschitz continuous gradient, $A\in \mathbb{R}^{m\times n_1}$, $B\in \mathbb{R}^{m\times n_2}$ and $C\in\mathbb{R}^{m\times n_3}$ are given matrixes, $b\in \mathbb{R}^{m}$ is a vector, with $f_1:\mathbb{R}^{n_1}\rightarrow \mathbb{R}\cup\{+\infty\}$ and $g_1:\mathbb{R}^{n_2}\rightarrow \mathbb{R}\cup\{+\infty\}$ are proper closed convex functions, $f_2:\mathbb{R}^{n_1}\rightarrow \mathbb{R}\cup\{+\infty\}$ and $g_2:\mathbb{R}^{n_2}\rightarrow \mathbb{R}\cup\{+\infty\}$ are continuous convex functions. We propose a majorized Bregman ADMM to solve the DC problem (1). Compared with the classical Bregman ADMM, the majorized Bregman ADMM only requires solving convex subproblems at each iteration, rather than DC subproblems. We prove that the sequence generated by the proposed method converges to a critical point of the augmented Lagrangian function, under the assumption that the potential function satisfies the Kurdyka-Łojasiewicz (KŁ) property. Preliminary numerical experiments are conducted to support our theoretical analysis.