A Majorized ADMM with Indefinite Proximal Terms for Linearly Constrained Convex Composite Optimization

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.