In this talk, we consider the problem of recovering a sparse vector from noisy measurement data. An algorithm of primal dual active set type for a class of sparsity-promoting penalties is proposed. A novel necessary optimality condition for the global minimizer using the associated thresholding operator is derived. The solutions to the optimality system are  coordinate-wise minimizers, and under minor conditions, they are also local minimizers. Upon introducing the dual variable, the active set can be determined from the primal and dual variables. This relation lends itself to an iterative algorithm of  active set type which at each step involves updating the primal variable only on the active set and then updating the dual variable explicitly.
Numerical experiments demonstrate its efficiency and accuracy.