Weighted iteration complexity of the sPADMM on the KKT residuals for convex composite optimization

TL;DR

This paper proves an $O(1/k)$ convergence rate for the sPADMM algorithm on convex composite problems, using a new generalized HPE formula, filling a gap in ergodic iteration complexity analysis.

Contribution

It introduces a novel generalized HPE iteration formula to establish the first $O(1/k)$ weighted iteration complexity for sPADMM on KKT residuals.

Findings

Establishes $O(1/k)$ weighted iteration complexity for sPADMM.

Fills the gap in ergodic iteration complexity analysis of ADMM variants.

Introduces a new generalized HPE iteration formula.

Abstract

In this paper we establish an $\mathcal{O}({1}/{k})$ weighted iteration complexity on the KKT residuals yielded by the sPADMM (semi-proximal alternating direction method of multiplier) for the convex composite optimization problem. This result, which is derived with the help of a novel generalized HPE (hybrid proximal extra-gradient) iteration formula, first fills the gap on the ergodic iteration complexity of the classic ADMM with a large step-size and its many proximal variants.