= Solution
At $x^k$, choose $a=f(x^k)$, $g=\nabla f(x^k)$, $H=\nabla^2f(x^k)$, 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 $\|s_k\|\leq\Delta_k$ and at least a fixed <Fraction of Cauchy decrease>. Its predicted decrease is then positive, so calculate the <trust-region acceptance ratio>
$$
\rho_k=\frac{f(x^k)-f(x^k+s_k)}{m_k(0)-m_k(s_k)}.
$$
Fix $0<\eta_1<\eta_2<1$, $0<\gamma_d<1<\gamma_u$, and a maximum radius $\Delta_{\max}$. One valid update is:
* If $\rho_k<\eta_1$, reject the trial: $x^{k+1}=x^k$ and $\Delta_{k+1}=\gamma_d\Delta_k$.
* Otherwise accept $x^{k+1}=x^k+s_k$. If additionally $\rho_k>\eta_2$ and the step reaches the boundary, enlarge the radius to $\min\{\gamma_u\Delta_k,\Delta_{\max}\}$; otherwise keep it unchanged.
The <trust region> controls the reliability of the model; the ratio decides whether that reliability was borne out by the actual <objective function>.
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