The linear convergence rate of the linearized version of alternating direction method of multipliers for convex optimization with three separable functions

Abstract

The linearized alternating direction methods of multipliers (L-ADMM) for solving convex minimization problems with two separable blocks in the objective functions is efficient. And its extended version (\(m \geqslant 3\)) is convergent under some mild conditions. Recently, The L-ADMM inspires much attention in analyzing its theoretical convergence rate. However, the research on its convergence rate is still in its infancy. In this paper, we consider the convergence rate of L-ADMM when solving the convex optimization problems that the subdifferentials of the underlying functions are piecewise linear multifunctions. Based on the error bound, we establish the linear convergence rate.