Structural conditions on trust-region radius updates. Part II: second-order convergence and asymptotic radius behaviour

arXiv:2610.26137v1 Announce Type: cross Abstract: Continuing Part I (arXiv:2606.30202), we ask not which structural conditions on $\{\Delta_k\}$ suffice for global convergence but what the radius does; the answers separate the mechanisms by whether the radius is tied to a criticality measure. We identify three asymptotic regimes: the criticality-anchored rules give $\sum_k \Delta_k^2/M_k \overline{\kappa}$ or $\overline{\mu} > \overline{\kappa}$, with $\overline{\kappa}$ a constant of the solution that cannot be checked in advance. These conditions are necessary, not artefacts: on $f(x) = \sin(x)/x$, $\overline{\kappa}$ is unbounded over the minimisers, so no parameter value meets the condition at all of them. Among the rules tied to a criticality measure, only those with an unbounded multiplier escape such a constant.