Barzilai--Borwein方法在R^d与希尔伯特空间中的尖锐最坏情况渐近速率
The Sharp Worst-Case Asymptotic Rate of the Barzilai--Borwein Method in $\mathbb R^d$ and Hilbert Spaces
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中文总结 AI 辅助
该研究建立了Barzilai--Borwein方法在有限维空间及希尔伯特空间中，针对一致正定二次函数与局部非线性问题的尖锐最坏情况渐近速率，明确了相关因子的边界与可达条件。
中文摘要 AI 辅助
我们针对两种Barzilai--Borwein（BB）规则在一致正定二次函数及局部非线性问题上建立了尖锐的渐近速率。在有限维空间中，对于任意固定规则和任意正初始步长，梯度根因子被限制为(b₀-a₀)/(b₀+a₀)，其中[a₀,b₀]是初始活跃谱区间，因此最坏轨迹因子为c_H=(κ(H)-1)/(κ(H)+1)。当H至少有两个不同特征值、采用匹配初始化和平衡端点轨迹时，可达到该值；在匹配初始化下，同一常数为最优一致包络阈值。对于希尔伯特空间上有界、自伴、一致正定算子，标量谱测度给出相应的活跃支撑界和最优匹配一致包络阈值，包括连续端点谱。最后，若梯度在驻点处严格Fréchet可微，且其导数为自伴、一致正定，则对于任意γ∈(c_*,1)（其中c_*=(κ(A_*)-1)/(κ(A_*)+1)），两种纯BB规则的局部速率均为一致包络；每个收敛到驻点的明确定义轨迹，其误差和梯度根因子至多为c_*，目标间隙根因子至多为c_*^2。在具有规定不同导数端点m_*<M_*的目标类中，R²上的二次函数与C^∞真正非二次示例的匹配端点轨迹可达到这些因子。
英文摘要
We establish sharp asymptotic rates for the two Barzilai--Borwein (BB) rules on uniformly positive quadratics and local nonlinear problems. In finite dimensions, for either fixed rule and an arbitrary positive first step, the gradient root factor is bounded by $(b_0-a_0)/(b_0+a_0)$, where $[a_0,b_0]$ is the initially active spectral interval. Hence the worst trajectory factor is $c_H=(κ(H)-1)/(κ(H)+1)$. When $H$ has at least two distinct eigenvalues, matched initialization and a balanced endpoint trajectory attain this value. Under matched initialization, the same constant is the optimal uniform-envelope threshold. For bounded, self-adjoint, uniformly positive operators on Hilbert space, scalar spectral measures yield the corresponding active-support bound and optimal matched uniform-envelope threshold, including continuous endpoint spectrum. Finally, if the gradient is strictly Fréchet differentiable at a stationary point and its derivative is self-adjoint and uniformly positive, every $γ\in(c_*,1)$, where $c_*=(κ(A_*)-1)/(κ(A_*)+1)$, is a uniform local envelope rate for either pure BB rule. Every well-defined trajectory converging to the stationary point has error and gradient root factors at most $c_*$ and objective-gap root factor at most $c_*^2$. Over the class of objectives with prescribed distinct derivative endpoints $m_*<M_*$, matched endpoint trajectories for quadratic and $C^\infty$ genuinely nonquadratic examples in $\mathbb R^2$ attain these factors.
