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

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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., x ∈ 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 O(1/k) ergodic iteration-complexity 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.