Sharpening Minimization Induced Penalties

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.