  We present a primal-dual algorithmic framework to obtain approximate
solutions to a prototypical constrained convex optimization problem, and
rigorously characterize how common structural assumptions affect the numerical
efficiency. Our main analysis technique provides a fresh perspective on
Nesterov's excessive gap technique in a structured fashion and unifies it with
smoothing and primal-dual methods. For instance, through the choices of a dual
smoothing strategy and a center point, our framework subsumes decomposition
algorithms, augmented Lagrangian as well as the alternating direction
method-of-multipliers methods as its special cases, and provides optimal
convergence rates on the primal objective residual as well as the primal
feasibility gap of the iterates for all.
