The Geometry of Monotone Operator Splitting Methods
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916487461076992 |
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| author | Combettes, Patrick L. |
| author_facet | Combettes, Patrick L. |
| contents | We propose a geometric framework to describe and analyze 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_08443 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Geometry of Monotone Operator Splitting Methods Combettes, Patrick L. Optimization and Control We propose a geometric framework to describe and analyze 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. |
| title | The Geometry of Monotone Operator Splitting Methods |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2310.08443 |