Total generalization variation (TGV) is a very powerful and important regularization for various inverse problems and computer vision tasks. In this paper, we proposed a semismooth Newton based augmented Lagrangian method to solve this problem. The augmented Lagrangian method (also called as method of multipliers) is widely used for lots of smooth or nonsmooth variational problems. However, its efficiency usually heavily depends on solving the coupled and nonlinear system together and simultaneously, which is very complicated and highly coupled for total generalization variation. With efficient primal-dual semismooth Newton methods for the complicated linear subproblems involving total generalized variation, we investigated a highly efficient and competitive algorithm compared to some efficient first-order method. 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.
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