  We study the convergence rates of the classical Lagrangian-based methods and
their variants for solving convex optimization problems with equality
constraints. We present a generalized prediction-correction framework to
establish $O(1/K^2)$ ergodic convergence rates. Under the strongly convex
assumption, based on the presented prediction-correction framework, some
Lagrangian-based methods with $O(1/K^2)$ ergodic convergence rates are
presented, such as the augmented Lagrangian method with the indefinite proximal
term, the alternating direction method of multipliers (ADMM) with a larger step
size up to $(1+\sqrt{5})/2$, the linearized ADMM with the indefinite proximal
term, and the multi-block ADMM type method (under an alternative assumption
that the gradient of one block is Lipschitz continuous).
