Work

$R$-Linear Convergence of Barzilai-Borwein Methods on Strictly Convex Quadratics via Closed-Cone Homogeneous Dynamics

Year: 2026

Type: preprint

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 $g\mapsto W^{1/2}g$ conjugates every fixed positive spectral-weight rule to BB1. Consequently, BB1, BB2, and all such weighted rules have the same homogeneous growth radius.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.12100

Authors: Yang, Shutai, Yuan, Ya-xiang

Institutions: University of Science and Technology of China, Chinese Academy of Sciences, Academy of Mathematics and Systems Science, State Key Laboratory of Scientific and Engineering Computing, University of Chinese Academy of Sciences

Language: English

Topic: Advanced Optimization Algorithms Research

Subfield: Numerical Analysis

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Natural Science Foundation of China, University of Science and Technology of China, National College Students Innovation and Entrepreneurship Training Program

Award: G8533620077

OpenAlex ID: W7212651442

Cites: 5

Cited by:

Related to: 10