Acceleration of saddle-point methods in smooth cases

TL;DR

This paper introduces a new convergence analysis for ADMM in smooth cases by linking it to oPDHG and proposes an accelerated ADMM variant with linear convergence, enhancing optimization efficiency.

Contribution

The paper provides a novel convergence analysis of ADMM via its equivalence with oPDHG and introduces an accelerated ADMM with proven linear convergence rate.

Findings

Accelerated ADMM converges linearly in smooth cases.

Equivalence between ADMM and oPDHG is established for analysis.

The proposed method improves convergence speed over standard ADMM.

Abstract

In the present paper we propose a novel convergence analysis of the Alternating Direction Methods of Multipliers (ADMM), based on its equivalence with the overrelaxed Primal-Dual Hybrid Gradient (oPDHG) algorithm. We consider the smooth case, which correspond to the cas where the objective function can be decomposed into one differentiable with Lipschitz continuous gradient part and one strongly convex part. An accelerated variant of the ADMM is also proposed, which is shown to converge linearly with same rate as the oPDHG.