  Motivated by the minimax concave penalty based variable selection in
high-dimensional linear regression, we introduce a simple scheme to construct
structured semiconvex sparsity promoting functions from convex sparsity
promoting functions and their Moreau envelopes. Properties of these functions
are developed by leveraging their structure. In particular, we provide sparsity
guarantees for the general family of functions. We further study the behavior
of the proximity operators of several special functions including indicator
functions of closed convex sets, piecewise quadratic functions, and the linear
combinations of them. To demonstrate these properties, several concrete
examples are presented and existing instances are featured as special cases.
