In order to remedy the underestimation of an advanced convex penalty defined via minimization, this paper proposes its nonconvex enhancement while preserving convexity of the associated regularized least-squares model. We first design a generalized Moreau enhanced minimization induced (GME-MI) penalty function by subtracting from the MI penalty its generalized Moreau envelope. Then, we derive an overall convexity condition for the GME-MI regularized least-squares model. Finally, under the overall convexity condition, characterizing the solution set of the GME-MI model with a carefully designed averaged nonexpansive operator, we develop a proximal splitting algorithm which is guaranteed to converge to a globally optimal solution. Numerical examples demonstrate the effectiveness of the proposed approach.

This paper presents a convex recovery method for block-sparse signals whose block partitions are unknown a priori. We first introduce a nonconvex penalty function, where the block partition is adapted for the signal of interest by minimizing the mixed l2/l1 norm over all possible block partitions. Then, by exploiting a variational representation of the l2 norm, we derive the proposed penalty function as a suitable convex relaxation of the nonconvex one. For a block-sparse recovery model designed with the proposed penalty, we develop an iterative algorithm which is guaranteed to converge to a globally optimal solution. Numerical experiments demonstrate the effectiveness of the proposed method.