  In this paper, we aim at unifying, simplifying and improving the convergence
rate analysis of Lagrangian-based methods for convex optimization problems. We
first introduce the notion of nice primal algorithmic map, which plays a
central role in the unification and in the simplification of the analysis of
most Lagrangian-based methods. Equipped with a nice primal algorithmic map, we
then introduce a versatile generic scheme, which allows for the design and
analysis of Faster LAGrangian (FLAG) methods with new provably sublinear rate
of convergence expressed in terms of function values and feasibility violation
of the original (non-ergodic) generated sequence. To demonstrate the power and
versatility of our approach and results, we show that most well-known iconic
Lagrangian-based schemes admit a nice primal algorithmic map, and hence share
the new faster rate of convergence results within their corresponding FLAG.
