PreprintCC BY 4.0arXiv
Global 𝒪(k−2) Convergence of Multilevel Gradient-Regularized Newton Methods
Abstract
We develop and analyze multilevel gradient-regularized Newton methods for unconstrained convex optimization with Lipschitz-continuous Hessians. By computing Newton corrections in lower-dimensional subspaces, the proposed methods aim to reduce computational costs while preserving the global 𝒪(k−2) convergence rate in function values of the full-space method. To establish this rate for multilevel methods, we introduce a class of row-nested restriction operators that ensures monotonicity of the reduced-gradient norms across levels. Numerical experiments with row-nested operators demonstrate convergence behavior comparable to full-space gradient-regularized Newton, while offering substantial computational savings on large-scale instances.
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BibTeX
@misc{tsipinakis2026global,
title = {{Global $\mathcalO\left(k^-2\right)$ Convergence of Multilevel Gradient-Regularized Newton Methods}},
author = {Nick Tsipinakis and Matthias Voigt},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35195},
eprint = {2610.35195},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35195},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Global 𝒪(k^−2) Convergence of Multilevel Gradient-Regularized Newton Methods
AU - Nick Tsipinakis
AU - Matthias Voigt
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35195
UR - https://arxiv.org/abs/2610.35195
AB - We develop and analyze multilevel gradient-regularized Newton methods for unconstrained convex optimization with Lipschitz-continuous Hessians. By computing Newton corrections in lower-dimensional subspaces, the proposed methods aim to reduce computational costs while preserving the global 𝒪(k^−2) convergence rate in function values of the full-space method. To establish this rate for multilevel methods, we introduce a class of row-nested restriction operators that ensures monotonicity of the reduced-gradient norms across levels. Numerical experiments with row-nested operators demonstrate convergence behavior comparable to full-space gradient-regularized Newton, while offering substantial computational savings on large-scale instances.
ER -