  In this paper, we consider a class of sparse group $\ell_0$ regularized
optimization problems. Firstly, we give a continuous relaxation model of the
considered problem and establish the equivalence of these two problems in the
sense of global minimizers. Then, we define a class of stationary points of the
relaxation problem, and prove that any defined stationary point is a local
minimizer of the considered sparse group $\ell_0$ regularized problem and
satisfies a desirable property of its global minimizers. Further, based on the
difference-of-convex (DC) structure of the relaxation problem, we design two DC
algorithms to solve the relaxation problem. We prove that any accumulation
point of the iterates generated by them is a stationary point of the relaxation
problem. In particular, all accumulation points have a common support set and a
unified lower bound for the nonzero entries, and their zero entries can be
attained within finite iterations. Moreover, we prove the convergence of the
entire iterates generated by the proposed algorithms. Finally, we give some
numerical experiments to show the efficiency of the proposed algorithms.
