  In this paper, we study a class of problems where the sum of truncated convex
functions is minimized. In statistical applications, they are commonly
encountered when $\ell_0$-penalized models are fitted and usually lead to
NP-Hard non-convex optimization problems. In this paper, we propose a general
algorithm for the global minimizer in low-dimensional settings. We also extend
the algorithm to high-dimensional settings, where an approximate solution can
be found efficiently. We introduce several applications where the sum of
truncated convex functions is used, compare our proposed algorithm with other
existing algorithms in simulation studies, and show its utility in
edge-preserving image restoration on real data.
