On the Linear Convergence Rate of Generalized ADMM for Convex Composite Programming

TL;DR

This paper proves the local linear convergence rate of a semi-proximal generalized ADMM method for convex composite optimization problems, enhancing understanding of its efficiency and global convergence properties.

Contribution

It establishes the local linear convergence rate of semi-proximal GADMM for linearly constrained convex problems, under calmness assumptions, and discusses its global convergence.

Findings

Proves local linear convergence rate of sPGADMM.

Provides inequalities for the generated sequence.

Discusses global convergence properties.

Abstract

Over the fast few years, the numerical success of the generalized alternating direction method of multipliers (GADMM) proposed by Eckstein \& Bertsekas [Math. Prog., 1992] has inspired intensive attention in analyzing its theoretical convergence properties. In this paper, we devote to establishing the linear convergence rate of the semi-proximal GADMM (sPGADMM) for solving linearly constrained convex composite optimization problems. The semi-proximal terms contained in each subproblem possess the abilities of handling with multi-block problems efficiently. We initially present some important inequalities for the sequence generated by the sPGADMM, and then establish the local linear convergence rate under the assumption of calmness. As a by-product, the global convergence property is also discussed.