Difference-of-Convex Algorithms for a Class of Sparse Group $\ell_0$ Regularized Optimization Problems

In this paper, we consider a class of sparse group $\ell_0$ regularized optimization problems. First, we give a continuous relaxation model of the considered problem and define a class of stationary points of the relaxation problem. Then, we establish the equivalence of these two problems in the sense of global minimizers, and prove that the defined stationary point is equivalent to the local minimizer of the considered sparse group $\ell_0$ regularized problem with a desirable bound from 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 local minimizer with a desirable bound for the considered sparse group $\ell_0$ problem. In particular, all accumulation points have a common support set and their zero entries can be attained within finite iterations. Moreover, we give the global convergence analysis of the proposed algorithms. Finally, we perform some numerical experiments to show the efficiency of the proposed algorithms.