 Solution

At $x^k$, choose $a=f(x^k)$, $g=\nabla f(x^k)$, $H={\nabla }^{2}f(x^k)$, and the current radius $\Delta ={\Delta }_k>0$. A bounded symmetric matrix approximation to the Hessian matrix can also be used. Stop if the gradient is sufficiently small; otherwise approximately solve the trust-region subproblem to obtain $s_k$ with $\Vert s_k\Vert \le {\Delta }_k$ and at least a fixed Fraction of Cauchy decrease. Its predicted decrease is then positive, so calculate the trust-region acceptance ratioFix $0<{\eta }_1<{\eta }_2<1$, $0<{\gamma }_d<1<{\gamma }_u$, and a maximum radius ${\Delta }_{\max }$. One valid update is:

The trust region controls the reliability of the model; the ratio decides whether that reliability was borne out by the actual objective function.