Initialization-dependent BFGS trial rates for tilted absolute values

Qiuyu Chen

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arXiv · 2609.09872

Initialization-dependent BFGS trial rates for tilted absolute values

Qiuyu Chen

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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.

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Qiuyu Chen. 2026-09-09. Initialization-dependent BFGS trial rates for tilted absolute values. https://arxiv.org/abs/2609.09872

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