The formulation min f(x)+g(y) subject to Ax+By=b, where f and g are extended-value convex functions, arises in many application areas such as signal processing, imaging and image processing, statistics, and machine learning either naturally or after variable splitting. In many common problems, one of the two objective functions is strictly convex and has Lipschitz continuous gradient. On this kind of problem, a very effective approach is the alternating direction method of multipliers (ADM or ADMM), which solves a sequence of f/g-decoupled subproblems. However, its effectiveness has not been matched by a provably fast rate of convergence; only sublinear rates such as O(1/k) and O(1/k^2) were recently established in the literature, though these rates do not require strict convexity. This paper shows that global linear convergence can be guaranteed under the above assumptions on strict convexity and Lipschitz gradient on one of the two functions, along with certain rank assumptions on A and B. The result applies to the generalized ADMs that allow the subproblems to be solved faster and less exactly in certain manners. In addition, the rate of convergence provides some theoretical guidance for optimizing the ADM parameters. In addition, this paper makes meaningful extensions to the existing global convergence theory of the generalized ADM.

## Citation

Rice University CAAM Technical Report TR12-14