Accelerated primal-dual methods with adaptive parameters for composite convex optimization with linear constraints

In this paper, we introduce two accelerated primal-dual methods tailored to address linearly constrained composite convex optimization problems, where the objective function is expressed as the sum of a possibly nondifferentiable function and a differentiable function with Lipschitz continuous gradient. The first method is the accelerated linearized augmented Lagrangian method (ALALM), which permits linearization to the differentiable function; the second method is the accelerated linearized proximal point algorithm (ALPPA), which enables linearization of both the differentiable function and the augmented term. By incorporating adaptive parameters, we demonstrate that ALALM achieves the $O(1/k^2)$ convergence rate and the linear convergence rate under the assumption of convexity and strong convexity, respectively. Additionally, we establish that ALPPA enjoys the $O(1/k)$ convergence rate in convex case and the $O(1/k^2)$ convergence rate in strongly convex case. We provide numerical results to validate the effectiveness of the proposed methods.