Generalized Symmetric Alternating Direction Method for Separable Convex Programming

The Alternating Direction Method of Multipliers (ADMM) has been proved to be effective for solving separable convex optimization subject to linear constraints. In this paper, we propose a Generalized Symmetric ADMM (GS-ADMM), which updates the Lagrange multiplier twice with suitable stepsizes, to solve the multi-block separable convex programming. This GS-ADMM partitions the data into two group variables so that one group consists of p block variables while the other has q block variables, where p ≥ 1 and q ≥ 1 are two integers. The two grouped variables are updated in a Gauss-Seidel fashion, and the blocks within each group are updated in a Jacobi scheme, which would make it very attractive for a big data setting. By adding proper proximal terms to the subproblems, we specify the domain of the stepsizes to guarantee that the GS-ADMM is globally convergent with a worst-case ergodic O(1/t) convergence rate. It turns out that our convergence domain of the stepsizes is significantly larger than other convergence domains in the literature. Hence, the GS-ADMM is more flexible on choosing and using larger stepsizes of the dual variable. Finally, two special cases of the GS-ADMM, which allows using zero penalty terms, are also discussed.