Relocated Fixed-Point Iterations with Applications to Variable Stepsize Resolvent Splitting
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arXiv
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| Format: | Preprint |
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2025
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| author | Atenas, Felipe Bauschke, Heinz H. Dao, Minh N. Tam, Matthew K. |
| author_facet | Atenas, Felipe Bauschke, Heinz H. Dao, Minh N. Tam, Matthew K. |
| contents | In this work, we develop a convergence framework for iterative algorithms whose updates can be described by a one-parameter family of nonexpansive operators. Within the framework, each step involving one of the main algorithmic operators is followed by a second step which ''relocates'' fixed-points of the current operator to the next. As a consequence, our analysis does not require the family of nonexpansive operators to have a common fixed-point, as is common in the literature. Our analysis uses a parametric extension of the demiclosedness principle for nonexpansive operators. As an application of our convergence results, we develop a version of the graph-based extension of the Douglas--Rachford algorithm for finding a zero of the sum of $N\geq 2$ maximally monotone operators, which does not require the resolvent parameter to be constant across iterations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_07428 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Relocated Fixed-Point Iterations with Applications to Variable Stepsize Resolvent Splitting Atenas, Felipe Bauschke, Heinz H. Dao, Minh N. Tam, Matthew K. Optimization and Control 47H05, 47H09, 47N10, 65K05, 47H04 In this work, we develop a convergence framework for iterative algorithms whose updates can be described by a one-parameter family of nonexpansive operators. Within the framework, each step involving one of the main algorithmic operators is followed by a second step which ''relocates'' fixed-points of the current operator to the next. As a consequence, our analysis does not require the family of nonexpansive operators to have a common fixed-point, as is common in the literature. Our analysis uses a parametric extension of the demiclosedness principle for nonexpansive operators. As an application of our convergence results, we develop a version of the graph-based extension of the Douglas--Rachford algorithm for finding a zero of the sum of $N\geq 2$ maximally monotone operators, which does not require the resolvent parameter to be constant across iterations. |
| title | Relocated Fixed-Point Iterations with Applications to Variable Stepsize Resolvent Splitting |
| topic | Optimization and Control 47H05, 47H09, 47N10, 65K05, 47H04 |
| url | https://arxiv.org/abs/2507.07428 |