A counterexample to a three-step convergence bound for BFGS with exact line search

Abstract

We disprove a three-step error estimate for the Broyden--Fletcher--Goldfarb--Shanno {(BFGS)} method with exact line searches. We construct a globally strongly convex function $F\in C^\infty(\mathbb R^3)$ and a single nonterminating trajectory $x_k\to0$ for which $\limsup_{k\to\infty}\|x_{k+3}\|/(\|x_{k+2}\|\|x_k\|)=+\infty$. The Hessian of $F$ lies between $\tfrac12 I$ and $\tfrac32 I$ everywhere, and $F(x)-\tfrac12\|x\|^2$ is compactly supported with all derivatives vanishing at $0$. The proof first constructs point-gradient pairs satisfying the exact BFGS step identities. A five-step cycle produces a large three-step quotient and restores an invariant plane for the actual updated matrix, allowing the cycle to be repeated with arbitrarily small gradient errors. Disjoint smooth corrections to a quadratic then realize all the pairs by one objective. Strong convexity identifies the prescribed points with the unique exact line-search minimizers. Thus smoothness and strong convexity do not imply the proposed product estimate, even for an objective that agrees with a quadratic to every order at its minimizer.