  By the asymptotic oracle property, non-convex penalties represented by
minimax concave penalty (MCP) and smoothly clipped absolute deviation (SCAD)
have attracted much attentions in high-dimensional data analysis, and have been
widely used in signal processing, image restoration, matrix estimation, etc.
However, in view of their non-convex and non-smooth characteristics, they are
computationally challenging. Almost all existing algorithms converge locally,
and the proper selection of initial values is crucial. Therefore, in actual
operation, they often combine a warm-starting technique to meet the rigid
requirement that the initial value must be sufficiently close to the optimal
solution of the corresponding problem. In this paper, based on the DC
(difference of convex functions) property of MCP and SCAD penalties, we aim to
design a global two-stage algorithm for the high-dimensional least squares
linear regression problems. A key idea for making the proposed algorithm to be
efficient is to use the primal dual active set with continuation (PDASC)
method, which is equivalent to the semi-smooth Newton (SSN) method, to solve
the corresponding sub-problems. Theoretically, we not only prove the global
convergence of the proposed algorithm, but also verify that the generated
iterative sequence converges to a d-stationary point. In terms of computational
performance, the abundant research of simulation and real data show that the
algorithm in this paper is superior to the latest SSN method and the classic
coordinate descent (CD) algorithm for solving non-convex penalized
high-dimensional linear regression problems.
