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In this paper, we consider the problem of recovering a sparse signal based on penalized least squares formulations. We develop a novel algorithm of primal-dual active set type for a class of nonconvex sparsity-promoting penalties, including ℓ^0, bridge, smoothly clipped absolute deviation, capped ℓ^1 and minimax concavity penalty. First we establish the existence of a global minimizer for the related optimization problems. Then we derive a novel necessary optimality condition for the globalarXiv:1310.1147v5 fatcat:keskxmncrrczbe2ksneb2kdl6m