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At the current point $x$, a trust-region method approximately minimizes a local quadratic modelwhere $B$ is a symmetric Hessian approximation and $\Delta >0$ is the trust-region radius. For a step with positive predicted decrease, compare the true and predicted reductions throughChoose fixed thresholds $0<{\eta }_1<{\eta }_2<1$. If $\rho <{\eta }_1$, the step is unsuccessful: reject it and reduce the radius. If ${\eta }_1\le \rho <{\eta }_2$, accept the successful step and normally retain the radius. If $\rho \ge {\eta }_2$, the step is very successful: accept it and permit an increase in radius, especially when the step reaches its boundary. The ratio tests whether the local model is accurate enough to justify moving. An approximate subproblem solver should deliver at least the model decrease achieved at the Cauchy point.

A trust-region method approximately minimizes a local model inside a bounded neighborhood, then tests the step using the trust-region acceptance ratio. Poor agreement rejects the trial and shrinks the radius; sufficient agreement accepts it; very good agreement may justify expanding the radius. A fraction of Cauchy decrease gives a quantitative minimum quality for approximate subproblem solves. The radius is an adaptive measure of the model's usable range, rather than a prescribed sequence tending to zero.