  In this paper we study nonconvex penalization using Bernstein functions whose
first-order derivatives are completely monotone. The Bernstein function can
induce a class of nonconvex penalty functions for high-dimensional sparse
estimation problems. We derive a thresholding function based on the Bernstein
penalty and discuss some important mathematical properties in sparsity
modeling. We show that a coordinate descent algorithm is especially appropriate
for regression problems penalized by the Bernstein function. We also consider
the application of the Bernstein penalty in classification problems and devise
a proximal alternating linearized minimization method. Based on theory of the
Kurdyka-Lojasiewicz inequality, we conduct convergence analysis of these
alternating iteration procedures. We particularly exemplify a family of
Bernstein nonconvex penalties based on a generalized Gamma measure and conduct
empirical analysis for this family.
