A counterexample to global convergence of classical DFP under the standard strong Wolfe conditions

What the math gives to ML

The paper gives a constructive failure mode for classical DFP that survives uniformly bounded objective curvature and standard strong-Wolfe line searches: the inverse-Hessian approximation can lose its smallest eigenvalue while its eigenspaces rotate indefinitely. This is valuable for neural-network quasi-Newton training because line-search acceptance and positive-definite updates alone do not prevent an optimizer from becoming effectively singular in changing gradient directions. The direct transfer is a spectrally safeguarded DFP optimizer that monitors inverse-Hessian conditioning and eigenspace rotation, then damps, clips, or resets problematic updates. The construction also supplies an adversarial benchmark: a smooth, globally well-conditioned two-dimensional objective on which Wolfe-compliant DFP fails despite benign true curvature.