Alternating direction method of multipliers for convex programming: a lift-and-permute scheme

TL;DR

This paper introduces a novel lift-and-permute scheme for the ADMM algorithm, unifying several existing methods and enabling accelerated convergence rates for linearly constrained convex optimization problems.

Contribution

It proposes a new lift-and-permute scheme that unifies various ADMM variants and achieves accelerated convergence rates for strongly convex objectives.

Findings

Unified framework for multiple ADMM variants

Achieves $O(1/k^2)$ convergence for strongly convex problems

Enhances efficiency of convex programming algorithms

Abstract

A lift-and-permute scheme of alternating direction method of multipliers (ADMM) is proposed for linearly constrained convex programming. It contains not only the newly developed balanced augmented Lagrangian method and its dual-primal variation, but also the proximal ADMM and Douglas-Rachford splitting algorithm. It helps to propose accelerated algorithms with worst-case $O(1/k^2)$ convergence rates in the case that the objective function to be minimized is strongly convex.