Linear Convergence Rate of Splitting Algorithms for Multi-Block Constrained Convex Minimizations

Multi-block linear constrained separable convex minimizations are ubiquitous and have been drawing increasing attention in recent researches. The alternating direction method of multipliers (ADMM) has been well studied and used in the literature for the two-block case. However, the direct extension of the ADMM to the multi-block case is not necessarily convergent. ADMM with Gaussian Back Substitution and ADMM with Prox-Parallel Splitting are two useful schemes to deal with the multi-block situation. Nevertheless, only a sublinear convergence rate was given in previous studies. In this paper, we prove the linear convergence rate of these two schemes under some assumptions. The proofs mainly depend on the variational inequalities.