  We introduce a novel primal-dual flow for affine constrained convex
optimization problem. As a modification of the standard saddle-point system,
our primal-dual flow is proved to possesses the exponential decay property, in
terms of a tailored Lyapunov function. Then a class of primal-dual methods for
the original optimization problem are obtained from numerical discretizations
of the continuous flow, and with a unified discrete Lyapunov function,
nonergodic convergence rates are established. Among those algorithms, we can
recover the (linearized) augmented Lagrangian method and the quadratic penalty
method with continuation technique. Also, new methods with a special inner
problem, that is a linear symmetric positive definite system or a nonlinear
equation which may be solved efficiently via the semi-smooth Newton method,
have been proposed as well. Especially, numerical tests on the linearly
constrained $l_1$-$l_2$ minimization show that our method outperforms the
accelerated linearized Bregman method.
