  This paper presents a majorized alternating direction method of multipliers
(ADMM) with indefinite proximal terms for solving linearly constrained
$2$-block convex composite optimization problems with each block in the
objective being the sum of a non-smooth convex function and a smooth convex
function, i.e., $\min_{x \in {\cal X}, \; y \in {\cal Y}}\{p(x)+f(x) +
q(y)+g(y)\mid A^* x+B^* y = c\}$.
  By choosing the indefinite proximal terms properly, we establish the global
convergence and $O(1/k)$ ergodic iteration-complexity of the proposed method
for the step-length $\tau \in (0, (1+\sqrt{5})/2)$. The computational benefit
of using indefinite proximal terms within the ADMM framework instead of the
current requirement of positive semidefinite ones is also demonstrated
numerically. This opens up a new way to improve the practical performance of
the ADMM and related methods.

Optimization and Control