  Lagrangian-based methods are classical methods for solving convex
optimization problems with equality constraints. We present novel
prediction-correction frameworks for such methods and their variants, which can
achieve $O(1/k)$ non-ergodic convergence rates for general convex optimization
and $O(1/k^2)$ non-ergodic convergence rates under the assumption that the
objective function is strongly convex or gradient Lipschitz continuous. We give
two approaches ($updating~multiplier~once$ $or~twice$) to design algorithms
satisfying the presented prediction-correction frameworks. As applications, we
establish non-ergodic convergence rates for some well-known Lagrangian-based
methods (esp., the ADMM type methods and the multi-block ADMM type methods).
