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In this paper we study decomposition methods based on separableapproximations for minimizing the augmented Lagrangian. In particular, we studyand compare the Diagonal Quadratic Approximation Method (DQAM) of Mulvey andRuszczy\'{n}ski and the Parallel Coordinate Descent Method (PCDM) ofRicht\'arik and Tak\'a\v{c}. We show that the two methods are equivalent forfeasibility problems up to the selection of a single step-size parameter.Furthermore, we prove an improved complexity bound for PCDM under strongconvexity, 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 degreeof partial separability and $L'$ and $\bar{L}$ are the maximum and average ofthe block Lipschitz constants of the gradient of the quadratic penaltyappearing in the augmented Lagrangian.

Separable Approximations and Decomposition Methods for the Augmented Lagrangian