{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:ZCPVE5VSXZCDVU3FZSP6QXX7ZI","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"8940d4dca655289693100266cd6e5f6870fb121dbd95952fb0d3c4d5dd065fdf","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.OC","submitted_at":"2026-05-12T19:16:13Z","title_canon_sha256":"8a4256f02ffcf7d9cbd40f404fe54aada1ecb1077600fd6b6ffe8e7cf4ee11ce"},"schema_version":"1.0","source":{"id":"2605.12666","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.12666","created_at":"2026-05-18T03:09:50Z"},{"alias_kind":"arxiv_version","alias_value":"2605.12666v1","created_at":"2026-05-18T03:09:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.12666","created_at":"2026-05-18T03:09:50Z"},{"alias_kind":"pith_short_12","alias_value":"ZCPVE5VSXZCD","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"ZCPVE5VSXZCDVU3F","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"ZCPVE5VS","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:00786d456527e6f74d0fc4de8341f9917adf022a75f5ab80383fc54eac4b6213","target":"graph","created_at":"2026-05-18T03:09:50Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"Under Lipschitz continuity of a preconditioned Hessian, the methods establish local superlinear and quadratic convergence guarantees, and the regularized variant attains an O(ε^{-3/2}) iteration complexity; an adaptive version preserves this while allowing inexact subproblem solutions."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"Existence of a nonlinear preconditioner such that the preconditioned Hessian satisfies Lipschitz continuity, and that a globalization strategy can be developed even when the preconditioned Newton direction is not necessarily a descent direction."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Nonlinear preconditioning extends Newton methods to objectives lacking Hessian Lipschitz continuity by analyzing a transformed mapping under a relaxed smoothness condition, with superlinear convergence and O(ε^{-3/2}) iteration complexity."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Newton methods achieve local superlinear and quadratic convergence by nonlinearly preconditioning the optimality mapping under Lipschitz continuity of the preconditioned Hessian."}],"snapshot_sha256":"1b18d12ce6c8c183e9a46f2ad6d58dee1b4af473ac0fd788116e462f476731c0"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"8d5e27429e83c2429c51fc63207074b2db4897dd24fca26969c6667e56f6b355"},"paper":{"abstract_excerpt":"Newton-type methods are typically analyzed under Lipschitz continuity of the Hessian, an assumption that can fail for objectives with higher-order or polynomial growth. We introduce a class of nonlinearly preconditioned Newton methods that apply Newton's root-finding scheme to a transformed optimality mapping, thereby extending recent nonlinear preconditioning ideas from first-order methods to the second-order setting. The resulting methods are naturally analyzed under Lipschitz continuity of a preconditioned Hessian, a condition that significantly relaxes the classical Hessian Lipschitz conti","authors_text":"Alexander Bodard, Panagiotis Patrinos","cross_cats":[],"headline":"Newton methods achieve local superlinear and quadratic convergence by nonlinearly preconditioning the optimality mapping under Lipschitz continuity of the preconditioned Hessian.","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.OC","submitted_at":"2026-05-12T19:16:13Z","title":"Newton methods beyond Hessian Lipschitz continuity: A nonlinear preconditioning approach"},"references":{"count":44,"internal_anchors":1,"resolved_work":44,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"Escaping saddle points without Lipschitz smoothness: the power of nonlinear preconditioning","work_id":"de72a345-424a-4149-827e-d87e9dbbcdc1","year":2025},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"Mirror and Preconditioned Gradient Descent in Wasserstein Space","work_id":"0135e5c4-cabc-4370-851a-0e2dc01ce976","year":2024},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"A generalized multivariable Newton method","work_id":"b0b73646-2017-4b8a-9781-2b7859336160","year":2021},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"A generalized univariate Newton method mo- tivated by proximal regularization","work_id":"04eab136-f5bf-4dfe-9bcb-808f0e6fa5a3","year":2012},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"Lower bounds for finding stationary points I","work_id":"78b7cdec-611a-4760-9df2-a0d435fc398b","year":2020}],"snapshot_sha256":"db90898c14634a973af045e018a4e29af33de4138b3a627c839374f340c95938"},"source":{"id":"2605.12666","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-14T20:27:59.597009Z","id":"230f1d47-f396-441f-9544-b2e574ed449a","model_set":{"reader":"grok-4.3"},"one_line_summary":"Nonlinear preconditioning extends Newton methods to objectives lacking Hessian Lipschitz continuity by analyzing a transformed mapping under a relaxed smoothness condition, with superlinear convergence and O(ε^{-3/2}) iteration complexity.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Newton methods achieve local superlinear and quadratic convergence by nonlinearly preconditioning the optimality mapping under Lipschitz continuity of the preconditioned Hessian.","strongest_claim":"Under Lipschitz continuity of a preconditioned Hessian, the methods establish local superlinear and quadratic convergence guarantees, and the regularized variant attains an O(ε^{-3/2}) iteration complexity; an adaptive version preserves this while allowing inexact subproblem solutions.","weakest_assumption":"Existence of a nonlinear preconditioner such that the preconditioned Hessian satisfies Lipschitz continuity, and that a globalization strategy can be developed even when the preconditioned Newton direction is not necessarily a descent direction."}},"verdict_id":"230f1d47-f396-441f-9544-b2e574ed449a"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:0ac43297d687b130472fce8116819b790f512cf79229392e94b311e13e2e7961","target":"record","created_at":"2026-05-18T03:09:50Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"8940d4dca655289693100266cd6e5f6870fb121dbd95952fb0d3c4d5dd065fdf","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.OC","submitted_at":"2026-05-12T19:16:13Z","title_canon_sha256":"8a4256f02ffcf7d9cbd40f404fe54aada1ecb1077600fd6b6ffe8e7cf4ee11ce"},"schema_version":"1.0","source":{"id":"2605.12666","kind":"arxiv","version":1}},"canonical_sha256":"c89f5276b2be443ad365cc9fe85effca13f7a647b1c772a244e3e6a8001331be","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c89f5276b2be443ad365cc9fe85effca13f7a647b1c772a244e3e6a8001331be","first_computed_at":"2026-05-18T03:09:50.329854Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:09:50.329854Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Xi21lE/wwwuZWZnQO4/BwRVhQpMgBQGOHQNvkKqti8K6nIerBBKbSzA9aKSeu53FNjM52mLa2dCl2f+Bvua4Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:09:50.330715Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.12666","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0ac43297d687b130472fce8116819b790f512cf79229392e94b311e13e2e7961","sha256:00786d456527e6f74d0fc4de8341f9917adf022a75f5ab80383fc54eac4b6213"],"state_sha256":"f3656d329e10896b1e94663383963d7d4124a78061451bc6df8e6dfb1fa8f3d0"}