## 3 Materials and Methods
The Hessian of this quadratic interpolation is \(\mathcal{O}(\Delta_{s})\). It was shown in [60] that the calculus-based approach to approximating gradients described in (1) and (3) is \(\mathcal{O}(\Delta_{s}^{2})\). It was also shown in [59; 60] that the calculus-based approach to approximating the Hessians described in (2) and (4) was \(\mathcal{O}(\Delta_{s})\). Hence, \(g^{k}\) and \(H^{k}\) are \(\mathcal{O}(\Delta^{k})\)-accurate. The next proposition shows that if \(g^{k}\) and \(H^{k}\) are both \(\mathcal{O}(\Delta^{k})\)-accurate approximations of the gradient and Hessian at \(x^{k}\), respectively, then the model function \(m^{k}\) is fully linear on \(B(x^{k};\Delta^{k})\). Proposition 1: _Let \(F\in\mathcal{C}^{2}\), \(x^{k}\in\operatorname{dom}F,s^{k}\in\mathbb{R}^{n}\) and \(\Delta^{k}>0\). Assume that \(x^{k}+s^{k}\in B(x^{k};\Delta^{k})\), where \(k\) is any iteration of Algorithm 1. Let the model function \(m^{k}\) be defined as in Equation (7). If \(g^{k}\) is an \(\mathcal{O}(\Delta^{k})\)-accurate approximation of the gradient of \(F\) at \(x^{k}\) and \(H^{k}\) is an \(\mathcal{O}(\Delta^{k})\)-accurate approximation of the Hessian of \(F\) at \(x^{k}\), then the model \(m^{k}\) is fully linear on \(B(x^{k};\Delta^{k})\)._  
```
Step 0: Initialization. Choose a feasible initial point \(x^{0}\), an initial trust-region radius \(\Delta^{0}>0\), an initial sampling radius \(0<\Delta_{s}^{0}\leq\Delta^{0}\), and a maximal trust-region radius \(\Delta_{\max}>0\). Build an initial fully linear model \(m^{0}(x^{0}+s)\) on \(B(x^{0};\Delta^{0})\). Denote by \(g^{0}\) and \(H^{0}\) the gradient and Hessian of the initial model at \(s=0\). Choose the constants \(0\leq\eta_{1}\leq\eta_{2}<1\) (with \(\eta_{2}\
eq 0\)), \(0<\gamma<1<\gamma_{inc},\epsilon_{stop}>0,\mu>0\). Set \(k=0\). fordo\(k=0,1,2,\ldots\) Step 1: Criticality step. If \(\|\tau_{m}^{k}\|\leq\epsilon_{stop}\) and \(\Delta^{k}\leq\mu\|\tau_{m}^{k}\|\),  stop. Return \(x^{k}\). If \(\|\tau_{m}^{k}\|\leq\epsilon_{stop}\) and \(\Delta^{k}>\mu\|\tau_{m}^{k}\|\),  set \(\Delta^{k}\leftarrow\min\{\mu\|\tau_{m}^{k}\|,\Delta^{k}\}\). If \(\Delta_{s}^{k}>\Delta^{k}\),  set \(\Delta_{s}^{k}\leftarrow\Delta^{k}\)  update the model \(m^{k}\) to make it fully linear on \(B(x^{k};\Delta^{k})\). Step 2: Trust-region sub-problem. Find an approximate solution \(s^{k}\) to the trust-region sub-problem (8). Step 3: Acceptance of the trial point. Compute \[\rho^{k}=\frac{F(x^{k})-F(x^{k}+s^{k})}{m^{k}(x^{k})-m^{k}(x^{k}+s^{k})}.\] If \(\rho^{k}\geq\eta_{1}\),  set \(x^{k+1}\gets x^{k}+s^{k}\) and build a fully linear model \(m^{k+1}\). Otherwise (\(\rho_{k}<\eta_{1}\)),  set \(m^{k+1}\gets m^{k}\) and \(x^{k+1}\gets x^{k}\). Step 4: Trust-region radius update. \[\Delta^{k+1}\in \begin{cases}[\Delta^{k},\min\{\gamma_{inc}\Delta^{k},\Delta_{ \max}\}],&\text{if}\quad\rho^{k}\geq\eta_{2},\\ \{\gamma\Delta^{k}\},&\text{if}\quad\rho^{k}<\eta_{2}.\end{cases} \] If \(\Delta_{s}^{k}>\Delta^{k}\),  set \(\Delta_{s}^{k+1}\leftarrow\Delta^{k}\). If \(\rho^{k}<\eta_{1}\),  attempt to improve the accuracy of the model. Increment \(k\) by one and go to Step 1. endfor
```  
**Algorithm 1****DFTR pseudo-code.**Proof. : For any \(x^{k}+s^{k}\in B(x^{k};\Delta^{k})\), we have  
\[\|\
abla F(x^{k}+s^{k})-\
abla m^{k}(x^{k}+s^{k})\| =\|\
abla F(x^{k}+s^{k})-g^{k}-(s^{k})^{\top}H^{k}\|\] \[\leq\|\
abla F(x^{k}+s^{k})-\
abla F(x^{k})+\
abla F(x^{k})-g^{k} \|+\Delta^{k}\|H^{k}\|\] \[\leq L_{g}\Delta^{k}+C_{1}\Delta^{k}+\|H^{k}\|\Delta^{k}\]  
where \(L_{g}\geq 0\) is the Lipschitz constant of \(\
abla F\) on \(B(x^{k};\Delta^{k})\), and \(C_{1}\geq 0\). Note that \(\|H^{k}\|\) is bounded above, since  
\[\|H^{k}\| =\|H^{k}-\
abla^{2}F(x^{k})+\
abla^{2}F(x^{k})\|\] \[\leq C_{2}\Delta^{k}+\|\
abla^{2}F(x^{k})\|.\]  
Since \(F\) is twice continuously differentiable on the box constraint, we have \(\|\
abla^{2}F(x^{k})\|\leq M\) for some nonnegative scalar \(M\) independently of \(k\). Therefore, \(\|H^{k}\|\leq C_{2}\Delta_{max}+M\).