{"paper":{"title":"An Enhanced Levenberg--Marquardt Method via Gram Reduction","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Chengchang Liu, John C.S. Lui, Luo Luo","submitted_at":"2024-12-11T17:27:30Z","abstract_excerpt":"This paper studied the problem of solving the system of nonlinear equations ${\\bf F}({\\bf x})={\\bf 0}$, where ${\\bf F}:{\\mathbb R}^{d}\\to{\\mathbb R}^d$. We propose Gram-Reduced Levenberg--Marquardt method which updates the Gram matrix ${\\bf J}(\\cdot)^\\top{\\bf J}(\\cdot)$ in every $m$ iterations, where ${\\bf J}(\\cdot)$ is the Jacobian of ${\\bf F}(\\cdot)$. Our method has a global convergence guarantee without relying on any step of line-search or solving sub-problems. We prove our method takes at most $\\mathcal{O}(m^2+m^{-0.5}\\epsilon^{-2.5})$ iterations to find an $\\epsilon$-stationary point of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.08561","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/2412.08561/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"}