Barzilai-Borwein Optimizer Provably Fails Superlinear Convergence on an Open Set of Quadratics for Every Dimension n≥4

arXiv 2607.21579·medium signal

Dawei Li, Xiaotian Jiang and Mingyi Hong prove a negative result for the Barzilai-Borwein step-size method: despite strong empirical performance in continuous optimization, it fails to converge superlinearly on an open set of quadratic problems in every dimension n≥4. An open set means the failure is not a measure-zero pathology — it survives perturbation. This bounds what practitioners can expect from BB-style adaptive step sizes and explains observed stagnation rather than treating it as a tuning problem.