\(O(1/k^2)\) convergence rates of (dual-primal) balanced augmented Lagrangian methods for linearly constrained convex programming

The recent balanced augmented Lagrangian method (ALM) and its dual-primal version are effective for solving linearly constrained convex programming problems. We present accelerated (dual-primal) balanced ALM methods and establish \(\varvec{O(1/k^2)}\) $\mathit{O}\mathbf{(}\mathbf{1}\mathbf{/}{\mathit{k}}^{\mathbf{2}}\mathbf{)}$ (where \(\varvec{k}\) $\mathit{k}$ is the number of iterations) convergence rates in the case that the objective function to be minimized is strongly convex. Numerical results demonstrate the efficiency of the new accelerated algorithms.