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

Abstract: Continuing Part I (arXiv:2606.30202), we ask not which structural conditions on ${Δ_k}$ 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 $\sum_k Δ_k^2/M_k &lt; +\infty$, hence $\liminf_k Δ_k = 0$, and $Δ_k \to 0$ once the model Hessians are bounded; the step-driven rules inherit the summability of the step series, hence $\sum_k Δ_k &lt; +\infty$; the fixed-factor rules keep $Δ_k$ bounded away from zero. Near a critical point with positive definite Hessian, convergence of a subsequence forces convergence of the full sequence. We then extend the framework to second-order criticality through $τ_k = \max{|g_k|, -λ_k<sup>-}$, obtaining convergence to second-order critical points with the worst-case complexity $$\mathcal{O}(\max{\varepsilon_g^{{-2}\varepsilon_H^{-1},} \varepsilon_H<sup>{-3}})$$, which reduces to $\mathcal{O}(\varepsilon<sup>{-3})$ for a common tolerance and is sharp for quadratic models. Finally we establish eventual inactivity of the constraint and superlinear convergence, isolating the two properties of the radius rule through which the mechanism enters; the rate itself is mechanism-independent. The criticality-anchored and capped gradient-scaled rules reach inactivity only under a side condition, $ζ&gt; \overlineκ$ or $\overlineμ &gt; \overlineκ$, with $\overlineκ$ a constant of the solution that cannot be checked in advance. These conditions are necessary, not artefacts: on $f(x) = \sin(x)/x$, $\overlineκ$ 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.