Construct a slowly growing cap for the gradient-scaled radius

Construct a cap for the gradient-scaled trust-region multiplier that grows slowly enough to exceed the solution-dependent threshold kbar eventually while ensuring that the radius Delta_k = mu_k omega_k still vanishes, thereby securing both the asymptotic radius properties and eventual trust-region inactivity.

Background

The paper studies gradient-scaled trust-region rules in which the radius is represented as Delta_k = mu_k omega_k, with omega_k a criticality measure. A finite multiplier cap supports the global asymptotic conclusions Delta_k to 0 and sum_k Delta_k2/M_k < +infty, which are important when the radius controls model or sampling accuracy.

In contrast, eventual inactivity near a strongly minimizing solution requires the multiplier to exceed a solution-dependent threshold kbar. The uncapped rule can achieve inactivity but may lose the global radius asymptotics when iterates do not converge to a strong minimizer; a fixed cap may preserve those asymptotics but fail to exceed kbar. The unresolved problem is therefore to design a cap schedule that achieves both properties simultaneously.

References

A cap growing slowly enough with $k$ to sit eventually above $\kbar$ while $\mu_k\omega_k$ still vanishes would secure both halves at once, and constructing one is the most concrete open problem this paper leaves.

— Structural conditions on trust-region radius updates. Part II: second-order convergence and asymptotic radius behaviour  (2610.26137 - Rieussec et al., 19 Oct 2026) in Section 8, paragraph "Rules that update the ratio"