A counterexample to global convergence of classical DFP under the standard strong Wolfe conditions
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 $n\ge 2$.