Identify alternative drivers for the radius-to-criticality ratio

Determine whether quantities other than the standard acceptance ratio rho_k and retrospective ratio widetilderho_{k+1} can drive updates of the radius-to-criticality ratio Delta_k/omega_k so as to produce the same eventual-inactivity effect.

Background

The analysis reduces eventual inactivity to whether the scalar ratio theta_k = Delta_k/|g_k| eventually exceeds a solution-dependent threshold. The gradient-scaled rule updates this ratio directly through its multiplier, while the retrospective gradient-scaled variant drives it using a retrospective acceptance ratio.

The paper leaves unresolved whether other algorithmic quantities could be used as drivers for the same ratio-adjustment mechanism. Such drivers could lead to new trust-region radius-update mechanisms with comparable inactivity guarantees.

References

Whether other quantities can drive the ratio to the same effect is open, and the family obtained by varying the driver looks to us the most promising direction for new mechanisms.

— 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"