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 (p(x) or q(y)) and a smooth convex function (f(x) or g(y)), i.e., minx∈X , y∈Y{p(x) + f(x) + q(y) + g(y) | A∗x + B∗y = c}. By choosing the indefinite proximal terms properly, we establish the global convergence, and the iteration-complexity in the non-ergodic sense of the proposed method for the step-length τ ∈ (0, (1 + √ 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.