On the convergence rate of the bi-alternating direction method of multipliers

In this paper, we analyze the convergence rate of the bi-alternating direction method of multipliers (BiADMM). Differently from ADMM that optimizes an augmented Lagrangian function, Bi-ADMM optimizes an augmented primal-dual Lagrangian function. The new function involves both the objective functions and their conjugates, thus incorporating more information of the objective functions than the augmented Lagrangian used in ADMM. We show that BiADMM has a convergence rate of O(K^-1) (K denotes the number of iterations) for general convex functions. We consider the lasso problem as an example application. Our experimental results show that BiADMM outperforms not only ADMM, but fast-ADMM as well.