{"paper":{"title":"Newton methods beyond Hessian Lipschitz continuity: A nonlinear preconditioning approach","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"Newton methods achieve local superlinear and quadratic convergence by nonlinearly preconditioning the optimality mapping under Lipschitz continuity of the preconditioned Hessian.","cross_cats":[],"primary_cat":"math.OC","authors_text":"Alexander Bodard, Panagiotis Patrinos","submitted_at":"2026-05-12T19:16:13Z","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"},"claims":{"count":4,"items":[{"kind":"strongest_claim","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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Newton methods achieve local superlinear and quadratic convergence by nonlinearly preconditioning the optimality mapping under Lipschitz continuity of the preconditioned Hessian.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"1b18d12ce6c8c183e9a46f2ad6d58dee1b4af473ac0fd788116e462f476731c0"},"source":{"id":"2605.12666","kind":"arxiv","version":1},"verdict":{"id":"230f1d47-f396-441f-9544-b2e574ed449a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T20:27:59.597009Z","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.","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","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.","pith_extraction_headline":"Newton methods achieve local superlinear and quadratic convergence by nonlinearly preconditioning the optimality mapping under Lipschitz continuity of the preconditioned Hessian."},"references":{"count":44,"sample":[{"doi":"","year":2025,"title":"Escaping saddle points without Lipschitz smoothness: the power of nonlinear preconditioning","work_id":"de72a345-424a-4149-827e-d87e9dbbcdc1","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Mirror and Preconditioned Gradient Descent in Wasserstein Space","work_id":"0135e5c4-cabc-4370-851a-0e2dc01ce976","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"A generalized multivariable Newton method","work_id":"b0b73646-2017-4b8a-9781-2b7859336160","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"A generalized univariate Newton method mo- tivated by proximal regularization","work_id":"04eab136-f5bf-4dfe-9bcb-808f0e6fa5a3","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"Lower bounds for finding stationary points I","work_id":"78b7cdec-611a-4760-9df2-a0d435fc398b","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":44,"snapshot_sha256":"db90898c14634a973af045e018a4e29af33de4138b3a627c839374f340c95938","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"8d5e27429e83c2429c51fc63207074b2db4897dd24fca26969c6667e56f6b355"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}