Initialization-dependent BFGS trial rates for tilted absolute values
Abstract
Lewis and Overton conjectured in 2008 that, for BFGS applied to the one-dimensional tilted absolute value with their Armijo--Wolfe line search, every nonterminating execution converges at a trial-normalized rate determined only by the tilt parameter and independent of the initial data. We prove that this initialization-independent rate assertion is false. In fact, for every tilt parameter in an explicit open interval, we construct two nonterminating executions with different sharp trial-normalized convergence rates. The proof reduces the line search to a scale-free state consisting of the iterate sign and the zero-crossing stepsize, and verifies two periodic state cycles by elementary rational inequalities. This yields an explicit family of counterexamples and shows that the asymptotic trial-normalized behavior of nonsmooth BFGS can depend essentially on the initialization even for this canonical one-dimensional model.
Related
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BibTeX
@misc{chen2026initializationdependent,
title = {{Initialization-dependent BFGS trial rates for tilted absolute values}},
author = {Qiuyu Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09872},
eprint = {2609.09872},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09872},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Initialization-dependent BFGS trial rates for tilted absolute values
AU - Qiuyu Chen
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09872
UR - https://arxiv.org/abs/2609.09872
AB - Lewis and Overton conjectured in 2008 that, for BFGS applied to the one-dimensional tilted absolute value with their Armijo--Wolfe line search, every nonterminating execution converges at a trial-normalized rate determined only by the tilt parameter and independent of the initial data. We prove that this initialization-independent rate assertion is false. In fact, for every tilt parameter in an explicit open interval, we construct two nonterminating executions with different sharp trial-normalized convergence rates. The proof reduces the line search to a scale-free state consisting of the iterate sign and the zero-crossing stepsize, and verifies two periodic state cycles by elementary rational inequalities. This yields an explicit family of counterexamples and shows that the asymptotic trial-normalized behavior of nonsmooth BFGS can depend essentially on the initialization even for this canonical one-dimensional model.
ER -