{"paper":{"title":"Q-quadratic convergence of the centralized circumcentered-reflection method under a relative interior condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The centralized circumcentered-reflection method converges Q-quadratically when sets share an affine hull, their relative interiors intersect, and relative boundaries are twice differentiable.","cross_cats":[],"primary_cat":"math.OC","authors_text":"Yunier Bello-Cruz","submitted_at":"2026-04-13T13:30:41Z","abstract_excerpt":"The centralized circumcentered-reflection method (\\cCRM) of Behling, Bello-Cruz, Iusem, and Santos~\\cite{Behling:2024} is known to converge superlinearly for the feasibility problem $\\operatorname{find}\\;z\\in X\\cap Y$ under a $\\mathcal{C}^1$ smoothness assumption on the boundaries of $X$ and $Y$. We sharpen this to a quantitative rate: when the boundaries are $\\mathcal{C}^2$ near the limit point $\\bar z$, \\cCRM\\ converges Q-quadratically, with an asymptotic constant \\(\n  2\\max(\\kappa_X,\\kappa_Y)/\\omega \\) governed by the boundary curvatures $\\kappa_X,\\kappa_Y$ at $\\bar z$ and the local error-b"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that cCRM converges superlinearly when aff(X)=aff(Y), ri(X)∩ri(Y)≠∅, and the relative boundaries are C^1 of appropriate relative dimension; and Q-quadratically when the relative boundaries are C^2, with explicit asymptotic constant expressed in terms of the boundary curvatures at the limit point and the local error-bound constant.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that aff(X)=aff(Y) and ri(X)∩ri(Y)≠∅ together with the relative boundaries being C^1 (or C^2) hypersurfaces of appropriate dimension; the paper explicitly leaves the case aff(X)≠aff(Y) as open.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"cCRM achieves Q-quadratic convergence to solutions of find z in X cap Y when aff(X)=aff(Y), ri(X) cap ri(Y) nonempty, and relative boundaries are C^2, with explicit rate constant from curvatures and local error bound.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The centralized circumcentered-reflection method converges Q-quadratically when sets share an affine hull, their relative interiors intersect, and relative boundaries are twice differentiable.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"fb86645c11768c88dbaab5115d09d93c623373c22d034a7a6e97d5e9fac27761"},"source":{"id":"2604.11450","kind":"arxiv","version":2},"verdict":{"id":"5df8d899-0d8c-4f60-a8d6-e5c611c9f631","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T15:49:16.571741Z","strongest_claim":"We prove that cCRM converges superlinearly when aff(X)=aff(Y), ri(X)∩ri(Y)≠∅, and the relative boundaries are C^1 of appropriate relative dimension; and Q-quadratically when the relative boundaries are C^2, with explicit asymptotic constant expressed in terms of the boundary curvatures at the limit point and the local error-bound constant.","one_line_summary":"cCRM achieves Q-quadratic convergence to solutions of find z in X cap Y when aff(X)=aff(Y), ri(X) cap ri(Y) nonempty, and relative boundaries are C^2, with explicit rate constant from curvatures and local error bound.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that aff(X)=aff(Y) and ri(X)∩ri(Y)≠∅ together with the relative boundaries being C^1 (or C^2) hypersurfaces of appropriate dimension; the paper explicitly leaves the case aff(X)≠aff(Y) as open.","pith_extraction_headline":"The centralized circumcentered-reflection method converges Q-quadratically when sets share an affine hull, their relative interiors intersect, and relative boundaries are twice differentiable."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.11450/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"}