SuperMann: a superlinearly convergent algorithm for finding fixed points of nonexpansive operators

TL;DR

SuperMann is a Newton-type algorithm that achieves superlinear convergence for fixed points of nonexpansive operators, improving robustness and accuracy over traditional methods in convex optimization.

Contribution

It introduces a novel separating hyperplane projection and generalizes Krasnosel'skii-Mann scheme, enabling superlinear convergence without requiring Jacobian nonsingularity.

Findings

SuperMann converges superlinearly under mild metric subregularity conditions.

It enhances robustness of operator splitting schemes against ill conditioning.

The method maintains the same oracle complexity as classical schemes.

Abstract

Operator splitting techniques have recently gained popularity in convex optimization problems arising in various control fields. Being fixed-point iterations of nonexpansive operators, such methods suffer many well known downsides, which include high sensitivity to ill conditioning and parameter selection, and consequent low accuracy and robustness. As universal solution we propose SuperMann, a Newton-type algorithm for finding fixed points of nonexpansive operators. It generalizes the classical Krasnosel'skii-Mann scheme, enjoys its favorable global convergence properties and requires exactly the same oracle. It is based on a novel separating hyperplane projection tailored for nonexpansive mappings which makes it possible to include steps along any direction. In particular, when the directions satisfy a Dennis-Mor\'e condition we show that SuperMann converges superlinearly under mild assumptions, which, surprisingly, do not entail nonsingularity of the Jacobian at the solution but merely metric subregularity. As a result, SuperMann enhances and robustifies all operator splitting schemes for structured convex optimization, overcoming their well known sensitivity to ill conditioning.