  We present in this paper first-order alternating linearization algorithms
based on an alternating direction augmented Lagrangian approach for minimizing
the sum of two convex functions. Our basic methods require at most
$O(1/\epsilon)$ iterations to obtain an $\epsilon$-optimal solution, while our
accelerated (i.e., fast) versions of them require at most
$O(1/\sqrt{\epsilon})$ iterations, with little change in the computational
effort required at each iteration. For both types of methods, we present one
algorithm that requires both functions to be smooth with Lipschitz continuous
gradients and one algorithm that needs only one of the functions to be so.
Algorithms in this paper are Gauss-Seidel type methods, in contrast to the ones
proposed by Goldfarb and Ma in [21] where the algorithms are Jacobi type
methods. Numerical results are reported to support our theoretical conclusions
and demonstrate the practical potential of our algorithms.
