AI 中文总结
本文针对任意强Wolfe参数构造了BFGS类方法在二维光滑非凸函数上发散的反例，证明其失效并非参数限制所致。
AI 中文摘要
自Dai（2002）提出里程碑式的反例以来，Broyden-Fletcher-Goldfarb-Shanno（BFGS）方法及其变体在现实线搜索参数下能否在光滑非凸函数上失效，一直是拟牛顿理论中的核心开放问题。Dai的反例仅限于小的Armijo参数（在Powell几何上$c_1 \le 1/84 \approx 0.0119$，或在其六点循环上$c_1 \le 69/7480 \approx 0.0092$），且目标函数无下界。本文解决了由Dai（2002）在与J. C. Gilbert讨论后提出的长期开放问题：对于每个Armijo参数$c_1 \in (0, 1)$，理论上是否存在这样的反例。具体地，对于任意给定的一对线搜索参数$0 < c_1 < c_2 < 1$，我们构造一个目标函数$f \in C^\infty(\mathbb{R}^2)$，有下界且梯度Lipschitz连续，使得广泛共轭类$\mathcal{C}$中配备首次局部极小点线搜索的每个方法都生成无穷迭代序列，满足对所有$k \ge 0$有$\\|\nabla f(x_k)\\| = 1$。类$\mathcal{C}$包含经典全内存BFGS方法、任意内存$m \ge 1$的有限内存BFGS（L-BFGS）方法、Broyden正定族以及Hestenes-Stiefel共轭梯度方法。步长是沿搜索射线的标准首次局部极小点，同时满足常数$(c_1, c_2)$的强Wolfe、弱Wolfe、Armijo和Goldstein条件。该构造在最小可能维度$n = 2$中运作，利用平面中的非衰减共轭下降轨道，结合显式管状插值，其双凸起轴向曲率剖面将Armijo比率置于$(0, 1)$中的任意位置。
英文摘要
Whether the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method and its variants can fail to converge on smooth nonconvex functions under realistic line-search parameters has remained a central open problem in quasi-Newton theory since the landmark counterexample of Dai (2002), which was confined to small Armijo parameters ($c_1 \le 1/84 \approx 0.0119$ on Powell's geometry, or $c_1 \le 69/7480 \approx 0.0092$ on his six-point cycle) and an objective function unbounded below. In this paper, we resolve the long-standing open question, posed by Dai (2002) following a discussion with J. C. Gilbert, of whether such counterexamples exist in theory for every Armijo parameter $c_1 \in (0, 1)$. Specifically, for every prescribed pair of line-search parameters $0 < c_1 < c_2 < 1$, we construct an objective function $f \in C^\infty(\mathbb{R}^2)$, bounded below and with Lipschitz continuous gradient, on which every method in a broad conjugacy class $\mathcal{C}$ equipped with the first-local-minimizer line search generates an infinite sequence of iterates with $\|\nabla f(x_k)\| = 1$ for all $k \ge 0$. The class $\mathcal{C}$ encompasses the classical full-memory BFGS method, limited-memory BFGS (L-BFGS) with arbitrary memory $m \ge 1$, the Broyden positive family, and the Hestenes-Stiefel conjugate gradient method. The steps are the standard first local minimizers along the search rays and simultaneously satisfy the strong Wolfe, weak Wolfe, Armijo, and Goldstein conditions with constants $(c_1, c_2)$. The construction operates in the minimal possible dimension $n = 2$, exploiting a non-decaying conjugate descent orbit in the plane coupled with an explicit tubular interpolation whose two-bump axial curvature profile places the Armijo ratio anywhere in $(0, 1)$.
Comments30 pages, 3 figures. Companion code: https://github.com/diaorui/bfgs-wolfe-counterexample
