发表机构
Academy of Mathematics and Systems Science, Chinese Academy of Sciences; University of Chinese Academy of Sciences; School of Mathematical Sciences, University of Science and Technology of China（中国科学院数学与系统科学研究院; 中国科学院大学; 中国科学技术大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过闭锥齐次动力学证明BB1、BB2及固定正谱权重变体在严格凸二次型上具有$R$-线性收敛，且所有规则共享相同的齐次增长半径。
AI 中文摘要
我们给出了BB1、BB2及固定正谱权重变体在有限维严格凸二次型上的$R$-线性收敛证明。延迟递推被写为相容连续梯度对闭锥上的一阶系统。其转移是连续的且正齐次度为1，包括在有限终止时。一个紧性定理表明，每个锥轨道的逐点收敛蕴含一致的有限步收缩，从而得到全局$R$-线性估计。通过一个实现引理，每个相容状态都启动一条BB1轨迹，因此Raydan定理在整锥上给出逐点稳定性。最后，变换$g\mapsto W^{1/2}g$将每个固定正谱权重规则共轭到BB1。因此，BB1、BB2及所有此类加权规则具有相同的齐次增长半径。
英文摘要
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.
Comments8 pages
