This paper is devoted to the numerical algorithms for solving the convex composite optimization. We present some new local convergence results for the linearized proximal algorithms with adaptive stepsizes (for short, ALP) proposed in Hu et al. (Appl Math Optim 87(52), 2023). Moreover, two new global versions of ALP are proposed by adding an evaluation step of the directions, and the corresponding global convergence results are established. The obtained local convergence results of this paper improve and extend the corresponding ones in Hu et al. (2023) by proving a higher convergence rate and relaxing the restrictions on the involved parameters. While our global convergence results fill the lack of global ones in Hu et al. (2023) for some more general problems.
Yang et al. (Wed,) studied this question.