  We address the minimization of a smooth objective function under an
$\ell_0$-constraint and simple convex constraints. When the problem has no
constraints except the $\ell_0$-constraint, some efficient algorithms are
available; for example, Proximal DC (Difference of Convex functions) Algorithm
(PDCA) repeatedly evaluates closed-form solutions of convex subproblems,
leading to a stationary point of the $\ell_0$-constrained problem. However,
when the problem has additional convex constraints, they become inefficient
because it is difficult to obtain closed-form solutions of the associated
subproblems. In this paper, we reformulate the problem by employing a new DC
representation of the $\ell_0$-constraint, so that PDCA can retain the
efficiency by reducing its subproblems to the projection operation onto a
convex set. Moreover, inspired by the Nesterov's acceleration technique for
proximal methods, we propose the Accelerated PDCA (APDCA), which attains the
optimal convergence rate if applied to convex programs, and performs well in
numerical experiments.
