PreprintCC BY-NC-ND 4.0arXiv
BFGS can converge to non-Clarke-stationary points with positive probability
Abstract
We construct a coercive, locally Lipschitz, semialgebraic function on ℝ2 for which BFGS with Armijo--Wolfe line search converges to a non-Clarke-stationary point from a nonempty open set of initial points and inverse-Hessian estimates. This disproves the almost-sure stationarity assertion in the 2013 challenge of Lewis and Overton. The construction and proof combine analytical arguments with computer-assisted verification of explicit inequalities.
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BibTeX
@misc{lai2026bfgs,
title = {{BFGS can converge to non-Clarke-stationary points with positive probability}},
author = {Lexiao Lai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34087},
eprint = {2610.34087},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34087},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - BFGS can converge to non-Clarke-stationary points with positive probability
AU - Lexiao Lai
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34087
UR - https://arxiv.org/abs/2610.34087
AB - We construct a coercive, locally Lipschitz, semialgebraic function on ℝ^2 for which BFGS with Armijo--Wolfe line search converges to a non-Clarke-stationary point from a nonempty open set of initial points and inverse-Hessian estimates. This disproves the almost-sure stationarity assertion in the 2013 challenge of Lewis and Overton. The construction and proof combine analytical arguments with computer-assisted verification of explicit inequalities.
ER -