{"ID":2895959,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.07428","arxiv_id":"2507.07428","title":"Relocated Fixed-Point Iterations with Applications to Variable Stepsize Resolvent Splitting","abstract":"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.","short_abstract":"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 opera...","url_abs":"https://arxiv.org/abs/2507.07428","url_pdf":"https://arxiv.org/pdf/2507.07428v2","authors":"[\"Felipe Atenas\",\"Heinz H. Bauschke\",\"Minh N. Dao\",\"Matthew K. Tam\"]","published":"2025-07-10T04:44:05Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}