发表机构
State Key Laboratory of Scientific and Engineering Computing, 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计算数学与科学工程计算国家重点实验室; 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
本文为二次 Barzilai-Borwein 方法构造显式 Lyapunov 函数，证明其 $R$-线性收敛，并给出端点指数的唯一极小极大性。
AI 中文摘要
Barzilai-Borwein (BB) 方法在强凸二次函数上是 $R$-线性收敛的，尽管目标函数值和梯度范数通常不是单调的。我们为预热步之后的二次动力学构造了一个显式的当前点 Lyapunov 函数，假设初始活动谱的两个端点在该步后仍然存在。若 $a<b$ 为这两个端点，$P_a,P_b$ 为相应的谱投影，则 \\[ \overline{f}(x):=\lVert P_a\nabla f(x)\rVert^{\frac{2b}{a+b}} \lVert P_b\nabla f(x)\rVert^{\frac{2a}{a+b}} \\] 对 BB1、BB2 以及每个固定的正加权延迟 Rayleigh 规则满足 \\[ \overline{f}(x_{k+1}) =\left(\frac{b-a}{b+a}\right)^2 e^{-2D_{a,b}(\tau_k)}\overline{f}(x_k), \qquad D_{a,b}(\tau_k)\ge0, \\] 其中 $\tau_k=1/\alpha_k$ 是步长的倒数。该函数通过对 Yang 和 Yuan 的尖锐速率分析中的端点余边界证书进行去归一化而得到。我们证明了端点指数是端点单项式中的唯一极小极大选择，并在一个非单调的 BB1 轨迹上说明了 Lyapunov 定律。利用该定律，我们还给出了该方法在有限维空间中 $R$-线性收敛的证明。
英文摘要
The Barzilai--Borwein (BB) method is $R$-linearly convergent on strongly convex quadratics, although neither the objective value nor the gradient norm is generally monotone. We construct an explicit current-point Lyapunov function for the quadratic dynamics after the warm-up step, assuming that both endpoints of the initial active spectrum survive this step. If $a<b$ are these endpoints and $P_a,P_b$ are the corresponding spectral projectors, then \[ \overline{f}(x) :=\lVert P_a\nabla f(x)\rVert^{\frac{2b}{a+b}} \lVert P_b\nabla f(x)\rVert^{\frac{2a}{a+b}} \] satisfies, for BB1, BB2, and every fixed positive weighted delayed Rayleigh rule, \[ \overline{f}(x_{k+1}) =\left(\frac{b-a}{b+a}\right)^2 e^{-2D_{a,b}(τ_k)}\overline{f}(x_k), \qquad D_{a,b}(τ_k)\ge0, \] where $τ_k=1/α_k$ is the reciprocal step. The function is obtained by denormalizing the endpoint coboundary certificate in the sharp-rate analysis of Yang and Yuan. We prove that the endpoint exponents are the unique minimax choice among endpoint monomials and illustrate the Lyapunov law on a nonmonotone BB1 trajectory. Using this law, we also give a proof of $R$-linear convergence of the method in finite dimensions.
Comments12 pages, 1 table
