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In this paper, we design an inertial accelerated primal-dual algorithm to address the convex-concave saddle point problem, which is formulated as $\min_{x}\max_{y} f(x) + \langle Kx, y \rangle - g(y)$. Remarkably, both functions $f$ and $g$ exhibit a composite structure, combining ``nonsmooth'' + ``smooth'' components. Under the assumption of partially strong convexity in the sense that $f$ is convex and $g$ is strongly convex, we introduce a novel inertial accelerated primal-dual algorithm based on Nesterov's extrapolation. This algorithm can be reduced to two classical accelerated forward-backward methods for unconstrained optimization problem. We show that the proposed algorithm achieves a non-ergodic $\mathcal{O}(1/k^2)$ convergence rate, where $k$ represents the number of iterations. Several numerical experiments validate the efficiency of our proposed algorithm.

The alternating direction method of multipliers (ADMM) has found widespread use in solving separable convex optimization problems. In this paper, by employing Nesterov extrapolation technique, we propose two families of accelerated linearized ADMMs for addressing two-block linearly constrained separable convex optimization problems where each block of the objective function exhibits a ``nonsmooth'' + ``smooth'' composite structure. Our proposed accelerated linearized ADMMs extend two classical Nesterov acceleration methods designed for unconstrained composite optimization problems to linearly constrained problems. These methods are capable of achieving non-ergodic convergence rates of $\mathcal{O}(1/k^2)$, provided that one block of the objective function exhibits strong convexity and the gradients of smooth terms are Lipschitz continuous. We show that the proposed methods can reduce to accelerated linearized augmented Lagrangian methods (ALMs) for solving one-block linearly constrained convex optimization problems. By choosing different extrapolation parameters, we explore the relationship between the proposed methods and some existing accelerated methods. Numerical results are presented to validate the efficacy and reliability of the proposed algorithms.