The \(\ell _1\) -regularized optimization problem has been extensively studied, leading to the development of various numerical algorithms. The active-set proximal-Newton algorithm, introduced in [J. Sci. Comput., 85(3):57, 2020], effectively solves box-constrained \(\ell _1\) optimization problems by distinguishing between active and free variables and integrating the proximal gradient method with Newton’s method to ensure iterative convergence. In this paper, we prove the superlinear or quadratic local convergence of the active-set proximal-Newton algorithm under a specific strict complementarity-like condition. For problems that do not satisfy this condition, we propose a two-stage active-set proximal-Newton algorithm and establish its superlinear or quadratic local convergence without the need for the strict complementarity-like condition. Numerical results on a set of test problems illustrate that the two-stage algorithm outperforms the original active-set proximal-Newton algorithm.
