This paper develops two general inertial smoothing Bregman proximal gradient algorithms to tackle the Capped- \(\ell _1\) group sparse regularization problem with a convex nonsmooth loss function. The first algorithm incorporates two distinct extrapolation terms into the gradient and proximal steps, with the parameters complying with certain constraints, while the other further employs a line search scheme in which the parameters are adaptively updated. We prove that any two accumulation points of the relevant subsequence share the same support set, and the zero groups can be exactly identified within a finite number of iterations. Moreover, any accumulation point is a lifted stationary point of the regularization model. Numerical experiments on several test problems demonstrate the efficiency of the proposed algorithms.
