Convergence study of indefinite proximal ADMM with a relaxation factor

Abstract

The alternating direction method of multipliers (ADMM) is widely used to solve separable convex programming problems. At each iteration, the classical ADMM solves the subproblems exactly. For many problems arising from practical applications, it is usually impossible or too expensive to obtain the exact solution of a subproblem. To overcome this, a special proximal term is added to ease the solvability of a subproblem. In the literature, the proximal term can be relaxed to be indefinite while still with a convergence guarantee; this relaxation permits the adoption of larger step sizes to solve the subproblem, which particularly accelerates its performance. A large value of the relaxation factor introduced in the dual step of ADMM also plays a vital role in accelerating its performance. However, it is still not clear whether these two acceleration strategies can be used simultaneously with no restriction on the penalty parameter. In this paper, we answer this question affirmatively and conduct a rigorous convergence analysis for indefinite proximal ADMM with a relaxation factor (IP-ADMM $$_{r}$$ ), reveal the relationships between the parameter in the indefinite proximal term and the relaxation factor to ensure its global convergence, and establish the worst-case convergence rate in the ergodic sense. Finally, some numerical results on basis pursuit and total variation-based denoising with box constraint problems are presented to verify the efficiency of IP-ADMM $$_{r}$$ .