BFGS for the unit ball

Let $g\in\mathbb{R}^n$ be an initial unit vector and let $H\in\mathbb{S}_{++}^n$ be an initial positive-definite matrix. Repeatedly define

$$s=-Hg,\qquad g_+=\frac{s}{\|s\|},\qquad y=g_+-g,\qquad V=I-\frac{sy^T}{s^Ty},\qquad H_+=VHV^T+\frac{ss^T}{s^Ty},$$

and then update $g=g_+$ and $H=H_+$.

BFGS for the unit ball. The trial step $s$ converges to zero.

This conjecture concerns the behavior of the BFGS update when the underlying compact convex set is the unit ball. The preceding discussion establishes determinant decrease for the update in the general membership algorithm, while the convergence of the trial steps in this special case is suggested only by numerical experiments.