Barzilai-Borwein Fails Superlinear Convergence on an Open Set of Quadratics for Every Dimension $n\geq 4$

2026-07-23

AI Summary

This study investigates the convergence rate of the Barzilai–Borwein (BB1) method for strictly convex quadratic optimization problems. For dimensions $n \geq 4$, the authors construct a counterexample and establish, for the first time, the existence of an open set with positive Lebesgue measure on which the BB1 method converges but fails to achieve superlinear convergence. By means of computer-assisted analysis of a projectivized dynamical system, they identify a non-resonant period-seven attracting orbit, revealing that both the gradient and the error exhibit two-sided geometric decay. Furthermore, they derive geometric lower bounds for the objective function gap, the gradient norm, and the energy norm of the error, rigorously ruling out superlinear convergence and thereby disproving the conjecture that BB1 converges superlinearly almost everywhere in high-dimensional settings.

Abstract

Barzilai--Borwein (BB) method has shown strong practical performance in continuous optimization, yet its convergence dynamics remains poorly understood. In particular, a central unresolved question is whether BB converges superlinearly for almost every strictly convex quadratic problem and initialization. We provide a negative answer to this question. Specifically, for every finite dimension $n\geq4$, we construct a nonempty open, hence positive-Lebesgue-measure, family of strictly convex quadratic problems and initial points for which the long Barzilai--Borwein method (BB1) converges but cannot converge root-superlinearly. More precisely, with the explicit constants $ρ_{\min}=10^{-6},ρ_{\max}=0.61$, every spectral component of the gradient is bounded above and below by the corresponding geometric sequence. Consequently, the gradient norm and the energy norm of the error satisfy two-sided geometric estimates with the same rates, while the objective gap satisfies the corresponding estimates with squared rates. In particular, all three quantities are bounded below by geometric sequences, ruling out superlinear convergence. The construction is highly nontrivial, based on a computer-assisted proof of a nonresonant, attracting seven-cycle of the projectivized BB dynamics in dimension four.

Problem

• Barzilai-Borwein
• superlinear convergence
• quadratic optimization
• convergence dynamics
• strictly convex

Innovation

• Barzilai-Borwein method
• superlinear convergence
• quadratic optimization
• computer-assisted proof
• dynamical systems