Mixed precision Newton's method for optimization
Abstract
Second-order optimization methods, such as Newton's algorithm, achieve fast local convergence and high accuracy, but their practical use is often limited by high computational costs. To mitigate this issue, variants such as inexact and quasi-Newton methods are widely used. A complementary and promising approach to improve the efficiency of the method is to employ mixed precision arithmetic, using different floating-point precisions for different operations, based on their impact on the convergence and accuracy of the method. In this work, we perform an error analysis of Newton's method accounting for different sources of inexactness, including approximations and rounding errors. We present a convergence analysis for the generated sequence, establishing bounds on the convergence rate and attainable accuracy. This theoretical framework covers quasi-Newton and inexact Newton methods, and is leveraged to propose mixed precision algorithms. We present a wide set of numerical experiments to illustrate our theoretical results and the behavior of Newton's method and its approximate variants in mixed precision floating-point arithmetic.
Related
- Convergence of a Randomized Newton Method in Nonconvex Optimization
- Envelope Cubic Regularized Newton Method for Nonconvex-Strongly-Concave Minimax Optimization
- A Globally Convergent Newton Method for Nonsmooth Optimization in the absence of Smooth Surrogates
- Fix-and-propagate heuristics using low-precision first-order LP solutions for large-scale mixed-integer linear optimization
- Acceleration of Newton's method for nonlinear algebraic systems by preflattening and elimination techniques
BibTeX
@misc{brisebarre2026mixed,
title = {{Mixed precision Newton's method for optimization}},
author = {Nicolas Brisebarre and Giuseppe Carrino and Theo Mary and Elisa Riccietti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.04828},
eprint = {2607.04828},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04828},
}
RIS
TY - UNPB
TI - Mixed precision Newton's method for optimization
AU - Nicolas Brisebarre
AU - Giuseppe Carrino
AU - Theo Mary
AU - Elisa Riccietti
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2607.04828
UR - https://arxiv.org/abs/2607.04828
AB - Second-order optimization methods, such as Newton's algorithm, achieve fast local convergence and high accuracy, but their practical use is often limited by high computational costs. To mitigate this issue, variants such as inexact and quasi-Newton methods are widely used. A complementary and promising approach to improve the efficiency of the method is to employ mixed precision arithmetic, using different floating-point precisions for different operations, based on their impact on the convergence and accuracy of the method. In this work, we perform an error analysis of Newton's method accounting for different sources of inexactness, including approximations and rounding errors. We present a convergence analysis for the generated sequence, establishing bounds on the convergence rate and attainable accuracy. This theoretical framework covers quasi-Newton and inexact Newton methods, and is leveraged to propose mixed precision algorithms. We present a wide set of numerical experiments to illustrate our theoretical results and the behavior of Newton's method and its approximate variants in mixed precision floating-point arithmetic.
ER -