{"paper":{"title":"A survey of trust-region radius update mechanisms. Part I: First-order analysis","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["cs.NA","math.NA"],"primary_cat":"math.OC","authors_text":"Fabian Bastin, J\\'er\\'emy Rieussec","submitted_at":"2026-06-29T12:21:32Z","abstract_excerpt":"We isolate three structural conditions on trust-region radius update rules for smooth unconstrained nonlinear optimisation, and study the class of mechanisms they define. The conditions act on the radius directly: a lower bound relative to the gradient norm, a contraction on unsuccessful iterations, and a controlled expansion on successful ones. A mechanism is \\emph{weakly admissible} if it satisfies the first two conditions, and \\emph{strongly admissible} if it satisfies the lower bound together with the controlled-expansion condition. Under uniformly bounded model Hessians, weak admissibilit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.30202","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/2606.30202/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"}