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In this paper, we consider the problem of recovering a sparse signal based on penalized least squares. We develop an algorithm of primal-dual active set type for a class of nonconvex sparsity-promoting penalties, which includes $\ell^0$, bridge, smoothly clipped absolute deviation, capped $\ell^1$ and minimax concavity penalty. First we establish the existence of a global minimizer for the class of optimization problems. Then we derive a novel necessary optimality condition for the global minimizer using the associated thresholding operator. 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. When combined with a continuation strategy on the regularization parameter, the primal dual active set method converges globally to the underlying regression target under some regularity conditions. Extensive numerical experiments demonstrate its superior performance in efficiency and accuracy compared with the existing methods.

A Unified Primal Dual Active Set Algorithm for Nonconvex Sparse Recovery