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The geometry of monotone operator splitting methods

Published online by Cambridge University Press:  04 September 2024

Patrick L. Combettes*
Affiliation:
North Carolina State University, Department of Mathematics, Raleigh, NC 27695-8205, USA E-mail: [email protected]
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Abstract

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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.

Type
Research Article
Copyright
© The Author(s), 2024. Published by Cambridge University Press

Footnotes

*

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

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