  Augmented Lagrangian method (also called as method of multipliers) is an
important and powerful optimization method for lots of smooth or nonsmooth
variational problems in modern signal processing, imaging, optimal control and
so on. However, one usually needs to solve the coupled and nonlinear system
together and simultaneously, which is very challenging. In this paper, we
proposed several semismooth Newton methods to solve the nonlinear subproblems
arising in image restoration, which leads to several highly efficient and
competitive algorithms for imaging processing. With the analysis of the metric
subregularities of the corresponding functions, we give both the global
convergence and local linear convergence rate for the proposed augmented
Lagrangian methods with semismooth Newton solvers.
