  In this paper we study decomposition methods based on separable
approximations for minimizing the augmented Lagrangian. In particular, we study
and compare the Diagonal Quadratic Approximation Method (DQAM) of Mulvey and
Ruszczy\'{n}ski and the Parallel Coordinate Descent Method (PCDM) of
Richt\'arik and Tak\'a\v{c}. We show that the two methods are equivalent for
feasibility problems up to the selection of a single step-size parameter.
Furthermore, we prove an improved complexity bound for PCDM under strong
convexity, and show that this bound is at least $8(L'/\bar{L})(\omega-1)^2$
times better than the best known bound for DQAM, where $\omega$ is the degree
of partial separability and $L'$ and $\bar{L}$ are the maximum and average of
the block Lipschitz constants of the gradient of the quadratic penalty
appearing in the augmented Lagrangian.
