Lagrangian-based methods in convex optimization: prediction-correction frameworks with ergodic convergence rates

Abstract: 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).