  Inspired by a second-order primal-dual dynamical system [Zeng X, Lei J, Chen
J. Dynamical primal-dual accelerated method with applications to network
optimization. 2019; arXiv:1912.03690], we propose an inertial primal-dual
method for the linear equality constrained convex optimization problem. When
the objective function has a "nonsmooth + smooth" composite structure, we
further propose an inexact inertial primal-dual method by linearizing the
smooth individual function and solving the subproblem inexactly. Assuming
merely convexity, we prove that the proposed methods enjoy $\mathcal{O}(1/k^2)$
convergence rate on $\mathcal{L}(x_k,\lambda^*)-\mathcal{L}(x^*,\lambda^*)$ and
$\mathcal{O}(1/k)$ convergence rate on primal feasibility, where $\mathcal{L}$
is the Lagrangian function and $(x^*,\lambda^*)$ is a saddle point of
$\mathcal{L}$. Numerical results are reported to demonstrate the validity of
the proposed methods.
