R-Linear Convergence of Barzilai-Borwein Methods on Strictly Convex Quadratics via Closed-Cone Homogeneous Dynamics
Abstract
We give an R-linear convergence proof for BB1, BB2, and fixed positive spectral-weight variants on finite-dimensional strictly convex quadratics. The delayed recurrence is written as a first-order system on the closed cone of compatible consecutive-gradient pairs. Its transition is continuous and positively homogeneous of degree one, including at finite termination. A compactness theorem shows that pointwise convergence of every cone orbit implies a uniform finite-step contraction and hence a global R-linear estimate. By a realization lemma, every compatible state initiates a BB1 trajectory, so Raydan's theorem yields pointwise stability on the entire cone. Finally, the transformation g↦ W1/2g conjugates every fixed positive spectral-weight rule to BB1. Consequently, BB1, BB2, and all such weighted rules have the same homogeneous growth radius.
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BibTeX
@misc{yang2026rlinear,
title = {{$R$-Linear Convergence of Barzilai-Borwein Methods on Strictly Convex Quadratics via Closed-Cone Homogeneous Dynamics}},
author = {Shutai Yang and Ya-xiang Yuan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12100},
eprint = {2609.12100},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12100},
}
RIS
TY - UNPB
TI - R-Linear Convergence of Barzilai-Borwein Methods on Strictly Convex Quadratics via Closed-Cone Homogeneous Dynamics
AU - Shutai Yang
AU - Ya-xiang Yuan
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.12100
UR - https://arxiv.org/abs/2609.12100
AB - We give an R-linear convergence proof for BB1, BB2, and fixed positive spectral-weight variants on finite-dimensional strictly convex quadratics. The delayed recurrence is written as a first-order system on the closed cone of compatible consecutive-gradient pairs. Its transition is continuous and positively homogeneous of degree one, including at finite termination. A compactness theorem shows that pointwise convergence of every cone orbit implies a uniform finite-step contraction and hence a global R-linear estimate. By a realization lemma, every compatible state initiates a BB1 trajectory, so Raydan's theorem yields pointwise stability on the entire cone. Finally, the transformation g↦ W^1/2g conjugates every fixed positive spectral-weight rule to BB1. Consequently, BB1, BB2, and all such weighted rules have the same homogeneous growth radius.
ER -