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The geometry of monotone operator splitting methods

Part of: Nonlinear operators and their properties

Published online by Cambridge University Press:  04 September 2024

Patrick L. Combettes Open the ORCID record for Patrick L. Combettes [Opens in a new window]
Patrick L. Combettes*
Affiliation:
North Carolina State University, Department of Mathematics, Raleigh, NC 27695-8205, USA E-mail: plc@math.ncsu.edu
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Abstract

We propose a geometric framework to describe and analyse a wide array of operator splitting methods for solving monotone inclusion problems. The initial inclusion problem, which typically involves several operators combined through monotonicity-preserving operations, is seldom solvable in its original form. We embed it in an auxiliary space, where it is associated with a surrogate monotone inclusion problem with a more tractable structure and which allows for easy recovery of solutions to the initial problem. The surrogate problem is solved by successive projections onto half-spaces containing its solution set. The outer approximation half-spaces are constructed by using the individual operators present in the model separately. This geometric framework is shown to encompass traditional methods as well as state-of-the-art asynchronous block-iterative algorithms, and its flexible structure provides a pattern to design new ones.

MSC classification

Primary: 47H05: Monotone operators and generalizations 90C48: Programming in abstract spaces 90C46: Optimality conditions, duality
Secondary: 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc. 90C25: Convex programming
Type
Research Article
Information
Acta Numerica , Volume 33 , July 2024 , pp. 487 - 632
Copyright
© The Author(s), 2024. Published by Cambridge University Press

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Footnotes

*

This work was supported by the National Science Foundation under grant CCF-2211123.

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