倾斜绝对值函数的初始化依赖BFGS试验率
Initialization-dependent BFGS trial rates for tilted absolute values
- Shanghai Jiao Tong University（上海交通大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文证明BFGS方法在一维倾斜绝对值函数上的收敛速率并非仅由倾斜参数决定，而是可能依赖于初始化，并构造了显式反例族。
中文摘要 AI 辅助
Lewis和Overton在2008年猜想，对于应用于一维倾斜绝对值函数并采用其Armijo-Wolfe线搜索的BFGS方法，每个非终止执行都以试验归一化速率收敛，该速率仅由倾斜参数决定，且与初始数据无关。我们证明这一初始化无关速率断言是错误的。事实上，对于显式开区间内的每个倾斜参数，我们构造了两个具有不同尖锐试验归一化收敛速率的非终止执行。证明将线搜索简化为由迭代符号和过零步长组成的无尺度状态，并通过初等有理不等式验证了两个周期状态循环。这产生了一个显式的反例族，并表明非光滑BFGS的渐近试验归一化行为即使对于这一典型的一维模型也可能本质上依赖于初始化。
英文摘要
Lewis and Overton conjectured in 2008 that, for BFGS applied to the one-dimensional tilted absolute value with their Armijo--Wolfe line search, every nonterminating execution converges at a trial-normalized rate determined only by the tilt parameter and independent of the initial data. We prove that this initialization-independent rate assertion is false. In fact, for every tilt parameter in an explicit open interval, we construct two nonterminating executions with different sharp trial-normalized convergence rates. The proof reduces the line search to a scale-free state consisting of the iterate sign and the zero-crossing stepsize, and verifies two periodic state cycles by elementary rational inequalities. This yields an explicit family of counterexamples and shows that the asymptotic trial-normalized behavior of nonsmooth BFGS can depend essentially on the initialization even for this canonical one-dimensional model.
