Generalized ADMM with optimal indefinite proximal term for linearly constrained convex optimization

We consider the generalized alternating direction method of multipliers (ADMM) for linearly constrained convex optimization. Many problems derived from practical applications have showed that usually one of the subproblems in the generalized ADMM is hard to solve, thus a special proximal term is added. In the literature, the proximal term can be indefinite which plays a vital role in accelerating numerical performance. In this paper, we are devoted to deriving the optimal lower bound of the proximal parameter and result in the generalized ADMM with optimal indefinite proximal term. The global convergence and the \begin{document}$ O(1/t) $\end{document} convergence rate measured by the iteration complexity of the proposed method are proved. Moreover, some preliminary numerical experiments on LASSO and total variation-based denoising problems are presented to demonstrate the efficiency of the proposed method and the advantage of the optimal lower bound.