Pith Number

pith:ZCPVE5VS

pith:2026:ZCPVE5VSXZCDVU3FZSP6QXX7ZI

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Newton methods beyond Hessian Lipschitz continuity: A nonlinear preconditioning approach

Alexander Bodard, Panagiotis Patrinos

Newton methods achieve local superlinear and quadratic convergence by nonlinearly preconditioning the optimality mapping under Lipschitz continuity of the preconditioned Hessian.

arxiv:2605.12666 v1 · 2026-05-12 · math.OC

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Claims

C1strongest 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.

C2weakest 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.

C3one 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.

References

44 extracted · 44 resolved · 1 Pith anchors

[1] Escaping saddle points without Lipschitz smoothness: the power of nonlinear preconditioning 2025

[2] Mirror and Preconditioned Gradient Descent in Wasserstein Space 2024

[3] A generalized multivariable Newton method 2021

[4] A generalized univariate Newton method mo- tivated by proximal regularization 2012

[5] Lower bounds for finding stationary points I 2020

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-18T03:09:50.329854Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c89f5276b2be443ad365cc9fe85effca13f7a647b1c772a244e3e6a8001331be

Aliases

arxiv: 2605.12666 · arxiv_version: 2605.12666v1 · doi: 10.48550/arxiv.2605.12666 · pith_short_12: ZCPVE5VSXZCD · pith_short_16: ZCPVE5VSXZCDVU3F · pith_short_8: ZCPVE5VS

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/ZCPVE5VSXZCDVU3FZSP6QXX7ZI \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: c89f5276b2be443ad365cc9fe85effca13f7a647b1c772a244e3e6a8001331be

Canonical record JSON

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    "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
  }
}