{"paper":{"title":"Linear convergence of the Douglas-Rachford algorithm via a generic error bound condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Javier Pe\\~na, Juan C. Vera, Luis F. Zuluaga","submitted_at":"2021-11-11T06:35:52Z","abstract_excerpt":"We provide new insight into the convergence properties of the Douglas-Rachford algorithm for the problem $\\min_x \\{f(x)+g(x)\\}$, where $f$ and $g$ are convex functions. Our approach relies on and highlights the natural primal-dual symmetry between the above problem and its Fenchel dual $\\min_{u} \\{ f^*(u) + g_*(u)\\}$ where $g_*(u):=g^*(-u)$. Our main development is to show the linear convergence of the algorithm when a natural error bound condition on the Douglas-Rachford operator holds. We leverage our error bound condition approach to show and estimate the algorithm's linear rate of converge"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2111.06071","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2111.06071/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}