This paper investigates the box-constrained \(\ell _0\) -regularized sparse optimization problem. We introduce the concept of \(\tau \) -stationary point and establish its connection to the local and global minimizers of the box-constrained \(\ell _0\) -regularized sparse optimization problem. We utilize the \(\tau \) -stationary points to define the support set, which we divide into active and inactive components. Subsequently, Newton’s method is employed to update the inactive variables, while the proximal gradient method is utilized to update the active variables. If Newton’s method fails, we use the proximal gradient method to update all variables. Under some mild conditions, we prove the global convergence and the local quadratic convergence rate. Finally, experimental results demonstrate the efficiency of our method.
