  This paper is concerned with a class of zero-norm regularized piecewise
linear-quadratic (PLQ) composite minimization problems, which covers the
zero-norm regularized $\ell_1$-loss minimization problem as a special case. For
this class of nonconvex nonsmooth problems, we show that its equivalent MPEC
reformulation is partially calm on the set of global optima and make use of
this property to derive a family of equivalent DC surrogates. Then, we propose
a proximal majorization-minimization (MM) method, a convex relaxation approach
not in the DC algorithm framework, for solving one of the DC surrogates which
is a semiconvex PLQ minimization problem involving three nonsmooth terms. For
this method, we establish its global convergence and linear rate of
convergence, and under suitable conditions show that the limit of the generated
sequence is not only a local optimum but also a good critical point in a
statistical sense. Numerical experiments are conducted with synthetic and real
data for the proximal MM method with the subproblems solved by a dual
semismooth Newton method to confirm our theoretical findings, and numerical
comparisons with a convergent indefinite-proximal ADMM for the partially
smoothed DC surrogate verify its superiority in the quality of solutions and
computing time.
