Asymptotic Analysis of an Abstract Stochastic Scheme for Solving Monotone Inclusions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908828455403520 |
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| author | Combettes, Patrick L. Madariaga, Javier I. |
| author_facet | Combettes, Patrick L. Madariaga, Javier I. |
| contents | We propose an abstract stochastic scheme for solving a broad range of monotone operator inclusion problems in Hilbert spaces. This framework allows for the introduction of stochasticity at several levels in monotone operator splitting methods: approximation of operators, selection of coordinates and operators in block-iterative implementations, and relaxation parameters. The analysis involves an abstract reduced inclusion model with two operators. At each iteration of the proposed scheme, stochastic approximations to points in the graphs of these two operators are used to form the update. The results are applied to derive the almost sure and $L^2$ convergence of stochastic versions of the proximal point algorithm, as well as of randomized block-iterative projective splitting methods for solving systems of coupled inclusions involving a mix of set-valued, cocoercive, and Lipschitzian monotone operators combined via various monotonicity-preserving operations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03023 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic Analysis of an Abstract Stochastic Scheme for Solving Monotone Inclusions Combettes, Patrick L. Madariaga, Javier I. Optimization and Control We propose an abstract stochastic scheme for solving a broad range of monotone operator inclusion problems in Hilbert spaces. This framework allows for the introduction of stochasticity at several levels in monotone operator splitting methods: approximation of operators, selection of coordinates and operators in block-iterative implementations, and relaxation parameters. The analysis involves an abstract reduced inclusion model with two operators. At each iteration of the proposed scheme, stochastic approximations to points in the graphs of these two operators are used to form the update. The results are applied to derive the almost sure and $L^2$ convergence of stochastic versions of the proximal point algorithm, as well as of randomized block-iterative projective splitting methods for solving systems of coupled inclusions involving a mix of set-valued, cocoercive, and Lipschitzian monotone operators combined via various monotonicity-preserving operations. |
| title | Asymptotic Analysis of an Abstract Stochastic Scheme for Solving Monotone Inclusions |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2512.03023 |