Separable approximations and decomposition methods for the augmented Lagrangian

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 Ruszczyński A diagonal quadratic approximation method for large scale linear programs, Oper. Res. Lett. 12 (1992), pp. 205–215 and the parallel coordinate descent method (PCDM) of Richtárik and Takáč Parallel coordinate descent methods for big data optimization. Technical report, November 2012. arXiv:1212.0873. We show that the two methods are equivalent for feasibility problems up to the selection of a 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′/L̄)(ω−1)2 times better than the best known bound for DQAM, where ω is the degree of partial separability and L’ and L̄ are the maximum and average of the block Lipschitz constants of the gradient of the quadratic penalty appearing in the augmented Lagrangian.