On convergence rates of proximal alternating direction method of multipliers

TL;DR

This paper analyzes the convergence rates of the proximal ADMM in Hilbert spaces, providing new theoretical results for both convex optimization and inverse problems, including linear and sublinear rates under regularity conditions.

Contribution

It offers novel non-ergodic convergence rate results for proximal ADMM in convex optimization and establishes noise-dependent convergence rates for inverse problems.

Findings

Improved non-ergodic convergence rates for convex problems.

Convergence rate results for inverse problems with noisy data.

Conditions under which linear and sublinear rates are achieved.

Abstract

In this paper we consider from two different aspects the proximal alternating direction method of multipliers (ADMM) in Hilbert spaces. We first consider the application of the proximal ADMM to solve well-posed linearly constrained two-block separable convex minimization problems in Hilbert spaces and obtain new and improved non-ergodic convergence rate results, including linear and sublinear rates under certain regularity conditions. We next consider proximal ADMM as a regularization method for solving linear ill-posed inverse problems in Hilbert spaces. When the data is corrupted by additive noise, we establish, under a benchmark source condition, a convergence rate result in terms of the noise level when the number of iteration is properly chosen.