A Linear Convergence Result for the Jacobi-Proximal Alternating Direction Method of Multipliers

In this paper, we analyze the convergence rate of the Jacobi-Proximal Alternating Direction Method of Multipliers (ADMM) initially introduced by Deng et al. for the block-structured optimization problem with linear constraint. The algorithm is well-suited for parallel implementation and widely used for large-scale multi-block optimization problems. While the o(1/k) convergence of the Jacobi-Proximal ADMM for the case $N \geq 3$ has been well-established in the previous work, to the best of our knowledge, its linear convergence for $N \geq 3$ remains unproven. We establish the linear convergence of the algorithm when the cost functions are strongly convex and smooth. Numerical experiments are presented supporting the convergence result.