$R$-Linear Convergence of Barzilai-Borwein Methods on Strictly Convex Quadratics via Closed-Cone Homogeneous Dynamics

We give an $R$-linear convergence proof for BB1, BB2, and fixed positive spectral-weight variants on finite-dimensional strictly convex quadratics. The delayed recurrence is written as a first-order system on the closed cone of compatible consecutive-gradient pairs. Its transition is continuous and positively homogeneous of degree one, including at finite termination. A compactness theorem shows that pointwise convergence of every cone orbit implies a uniform finite-step contraction and hence a global $R$-linear estimate. By a realization lemma, every compatible state initiates a BB1 trajectory, so Raydan’s theorem yields pointwise stability on the entire cone. Finally, the transformation $g\mapsto W^{1\mathrm{/}2}g$ conjugates every fixed positive spectral-weight rule to BB1. Consequently, BB1, BB2, and all such weighted rules have the same homogeneous growth radius.