AI 中文总结
该研究构造反例证明经典DFP方法在标准强沃尔夫条件下无法全局收敛，突破了拟牛顿优化领域的长期开放性问题。
AI 中文摘要
拟牛顿优化领域有一个长期未决的开放性问题：当所有接受的步长都满足标准弱沃尔夫条件时，经典Davidon–Fletcher–Powell（DFP）方法是否在一致凸目标函数上全局收敛？我们证明答案是否定的，即使在标准强沃尔夫条件下也不成立。固定0<c₁<2/3且2/3≤c₂<1，我们构造一个属于C²(ℝ²)的函数f，使得对所有x∈ℝ²，都有(1/2)I≼∇²f(x)≼(3/2)I；同时选择固定的正定初始逆海森近似和正步长序列，经典DFP迭代是良定的，所有接受的步长均满足标准强沃尔夫条件，但|∇f(xₖ)|收敛到一个正常数，全局海森条件数至多为3。该构造利用了一维不变中心流形附近的两步交替DFP序列，沿此序列逆海森近似的较小特征值趋于0，循环起点间的梯度范数变化是可和的，但相关特征向量的总旋转无界，DFP序列的聚点构成一个圆；均匀分离界允许我们插值规定的函数值和梯度，我们将具有两两不交支撑的光滑函数添加到二次函数中并保持全局海森界，仿射变量变换得到初始单位矩阵的示例，其海森界依赖于问题，正交直和将结果推广到所有n≥2的维度。
英文摘要
A long-standing open question in quasi-Newton optimization asks whether the classical Davidon--Fletcher--Powell (DFP) method converges globally on uniformly convex objectives when all accepted steps satisfy the standard weak Wolfe conditions. We show that the answer is no, even under the standard strong Wolfe conditions. Fix $0<c_1<2/3$ and $2/3\le c_2<1$. We construct a function $f\in C^2(\mathbb{R}^2)$ such that $\frac{1}{2}I\preceq\nabla^2 f(x)\preceq\frac{3}{2}I$ for all $x\in\mathbb{R}^2$. We also choose a fixed positive definite initial inverse Hessian approximation and a sequence of positive step lengths. The classical DFP iteration is well defined, and all accepted steps satisfy the standard strong Wolfe conditions, but $|\nabla f(x_k)|$ converges to a positive constant. The global Hessian condition number is at most three. The construction uses an alternating two-step DFP sequence near a one-dimensional invariant center manifold. Along this sequence, the smaller eigenvalue of the inverse Hessian approximation tends to zero. The changes in the gradient norm between cycle starts are summable, but the total rotation of the associated eigenvectors is unbounded. The accumulation points of the DFP sequence form a circle. A uniform separation bound allows us to interpolate the prescribed function values and gradients. We add smooth functions with pairwise disjoint supports to a quadratic and keep the global Hessian bounds. An affine change of variables gives an identity-initialized example with problem-dependent Hessian bounds. An orthogonal direct sum extends the result to every dimension $n\ge 2$.
Comments28 pages, 2 figures, 2 tables
