Preconditioned ADMM with nonlinear operator constraint

TL;DR

This paper introduces a preconditioned ADMM algorithm tailored for convex optimization problems with nonlinear constraints, demonstrating its effectiveness in nonlinear MRI reconstruction and connecting it to NL-PDHGM.

Contribution

It proposes a novel preconditioned ADMM variant for nonlinear constraints and links it to NL-PDHGM, advancing nonlinear inverse problem solving.

Findings

Successfully applied to nonlinear MRI reconstruction

Shows improved convergence properties

Establishes theoretical connections to NL-PDHGM

Abstract

We are presenting a modification of the well-known Alternating Direction Method of Multipliers (ADMM) algorithm with additional preconditioning that aims at solving convex optimisation problems with nonlinear operator constraints. Connections to the recently developed Nonlinear Primal-Dual Hybrid Gradient Method (NL-PDHGM) are presented, and the algorithm is demonstrated to handle the nonlinear inverse problem of parallel Magnetic Resonance Imaging (MRI).