  Recently, several nonconvex sparse regularizers which can preserve the
convexity of the cost function have received increasing attention. This paper
proposes a general class of such convexity-preserving (CP) regularizers, termed
partially smoothed difference-of-convex (pSDC) regularizer. The pSDC
regularizer is formulated as a structured difference-of-convex (DC) function,
where the landscape of the subtrahend function can be adjusted by a
parameterized smoothing function so as to attain overall-convexity. Assigned
with proper building blocks, the pSDC regularizer reproduces existing CP
regularizers and opens the way to a large number of promising new ones.
  With respect to the resultant nonconvexly regularized convex (NRC) model, we
derive a series of overall-convexity conditions which naturally embrace the
conditions in previous works. Moreover, we develop a unified framework based on
DC programming for solving the NRC model. Compared to previously reported
proximal splitting type approaches, the proposed framework makes less stringent
assumptions. We establish the convergence of the proposed framework to a global
minimizer. Numerical experiments demonstrate the power of the pSDC regularizers
and the efficiency of the proposed DC algorithm.
