This work focuses on an unconstrained convex optimization problem with a “smooth”+“nonsmooth” composite structure. We propose an accelerated forward-backward algorithm with subgradient correction, derived via time discretization of an inertial dynamical system featuring vanishing damping \(\alpha /t\) and Hessian-driven damping, as studied by Attouch et al. (Math. Program. 193, 113–155, 2022). For \(\alpha \ge 3\) , the algorithm achieves an \({\mathcal {O}}(1/k^2)\) convergence rate for the objective residual, along with an inverse cubic rate for the squared subdifferential norm. Furthermore, when \(\alpha > 3\) , we establish the convergence of the iterative sequence and improve the convergence rate of the objective residual to \(o(1/k^2)\) . Additionally, we explore an inexact variant of the proposed algorithm, where the proximal subproblem is solved approximately. Under mild assumptions on the error sequences, we show that the fast convergence properties are retained. Numerical experiments are provided to demonstrate the effectiveness and robustness of the proposed methods.
