SuperMann: A Superlinearly Convergent Algorithm for Finding Fixed Points of Nonexpansive Operators

Andreas Themelis, Panagiotis Patrinos

Research output: Contribution to journalArticlepeer-review

16 Citations (Scopus)


Operator splitting techniques have recently gained popularity in convex optimization problems arising in various control fields. Being fixed-point iterations of nonexpansive (NE) 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 NE 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-Moré 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.

Original languageEnglish
Article number8675506
Pages (from-to)4875-4890
Number of pages16
JournalIEEE Transactions on Automatic Control
Issue number12
Publication statusPublished - Dec 2019
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering


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