The convergence rate of the proximal alternating direction method of multipliers with indefinite proximal regularization.

The proximal alternating direction method of multipliers (P-ADMM) is an efficient first-order method for solving the separable convex minimization problems. Recently, He et al. have further studied the P-ADMM and relaxed the proximal regularization matrix of its second subproblem to be indefinite. This is especially significant in practical applications since the indefinite proximal matrix can result in a larger step size for the corresponding subproblem and thus can often accelerate the overall convergence speed of the P-ADMM. In this paper, without the assumptions that the feasible set of the studied problem is bounded or the objective function's component $\theta_{i}(\cdot)$ of the studied problem is strongly convex, we prove the worst-case $\mathcal{O}(1/t)$ convergence rate in an ergodic sense of the P-ADMM with a general Glowinski relaxation factor $\gamma\in(0,\frac{1+\sqrt{5}}{2})$ , which is a supplement of the previously known results in this area. Furthermore, some numerical results on compressive sensing are reported to illustrate the effectiveness of the P-ADMM with indefinite proximal regularization. [ABSTRACT FROM AUTHOR]