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

arXiv: https://arxiv.org/abs/1609.06955 · HTML: https://mlsanity.com/paper/1609.06955/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Sep 2016.

## Built on by (most cited first, 4 of 4)

- [Robust Online Learning over Networks](https://mlsanity.com/paper/2309.00520/) (2023)
- [Learning to accelerate Krasnosel'skii-Mann fixed-point iterations with guarantees](https://mlsanity.com/paper/2601.07665/) (2026)
- [SPIRAL: A superlinearly convergent incremental proximal algorithm for nonconvex finite sum minimization](https://mlsanity.com/paper/2207.08195/) (2022)
- [A regret minimization approach to fixed-point iterations](https://mlsanity.com/paper/2509.21653/) (2025)

## Abstract (arXiv)

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.