Block-Wise ADMM with a Relaxation Factor for Multiple-Block Convex Programming

It has been shown that the alternating direction method of multipliers(ADMM)is not necessarily convergent when it is directly extended to a multiple-block linearly constrained convex minimization model with an objective function that is in the sum of more than two functions without coupled variables.Recently,we pro-posed the block-wise ADMM,which was obtained by regrouping the variables and functions of such a model as two blocks and then applying the original ADMM in block-wise.This note is a further study on this topic with the purpose of showing that a well-known relaxation factor proposed by Fortin and Glowinski for iteratively updat-ing the Lagrangian multiplier of the originalADMMcan also be used in the block-wise ADMM.We thus propose the block-wise ADMM with Fortin and Glowinski’s relax-ation factor for the multiple-block convex minimization model.Like the block-wise ADMM,we also suggest further decomposing the resulting subproblems and regular-izing them by proximal terms to ensure the convergence.For the block-wise ADMM with Fortin and Glowinski's relaxation factor,its convergence and worst-case conver-gence rate measured by the iteration complexity in the ergodic sense are derived.