An inexact symmetric ADMM algorithm with indefinite proximal term for sparse signal recovery and image restoration problems

Compared with the alternating direction method of multipliers (ADMM), the symmetric ADMM, which updates the Lagrange multiplier twice in each iteration, is a more efficient approach for solving linearly constrained convex optimization problems. However, the difficulty of solving subproblems has a central role in practical applications. In this paper, we develop an inexact symmetric ADMM with an indefinite proximal term for linearly constrained convex optimization problems. To the best of our knowledge, this is the first variant of the ADMM that unifies the relative error criteria and indefinite proximal term. Specifically, both subproblems in the proposed algorithm can be approximately solved by certain relative error criteria. Moreover, the proximal term in the second subproblem is allowed to be indefinite while still theoretically guaranteeing the convergence. We establish its global convergence and worst-case O ( 1 / N ) convergence rate in the ergodic sense. We apply the new method to solve ℓ 1 regularized analysis sparse recovery and constrained TV- ℓ 2 image restoration problems, and some numerical results are reported to verify the efficiency of the proposed algorithm.