## 2 Materials and Methods
### Non-Linear Least Square Fitting Algorithm
The scaled Cauchy step \(s_{k}^{C}\) is defined by solving the constrained minimization problem inside the trust region within the box-boundary where the step \(s\) is constrained to lie on the direction given by the scaled gradient \(-D_{k}\
abla f_{k}\)  
\[s_{k}^{C}=\operatorname*{argmin}_{s\in\text{span}\{-D_{k}\
abla f_{k}\}}m_{k} (s),\quad\text{with}\ \|s\|_{2}\leq\Delta_{k},\quad x_{k}+s\in[l,u]. \tag{13}\]  
Since the solution of the problem (13) is of the form \(q_{k}=-c\ D_{k}\
abla f_{k},\quad c\in\mathbb{R}\)[18], the initial \(q_{k}\) is found being  
\[q_{k}=-\left(\frac{\left\|D_{k}^{\frac{1}{k}}\
abla f_{k}\right\|_{2}}{\left\| hD_{k}\
abla f_{k}\right\|_{2}}\right)^{2}D_{k}\
abla f_{k}, \tag{14}\]  
ensuring it remains within the box constraints \(l,u\). If \(q_{k}\) lies outside the current trust region, it is adjusted to stay on the trust region boundary by normalizing by the norm of the gradient:  
\[x_{k+1}=x_{k}+s_{k},\quad\forall k\in\mathbb{N}. \tag{6}\]0  
To ensure the step is feasible, the maximum possible step length before meeting the box bound in the current gradient direction is computed, where \(\alpha\) is defined as  
\[\alpha=\min_{1\leq i\leq n}\alpha_{i  
leading to  
\[s_{k}^{C}= \begin{cases}q_{k}&\text{if }x_{k}+q_{k}\in[l,u]\\ \alpha q_{k},&\text{otherwise}\end{cases} , \tag{17}\]  
Then, in the case that \(s_{k}^{N}\) from (10) is outside the trust region, a dogleg step between \(s_{k}^{N}\) and the Cauchy direction is computed. The classical Cauchy step \(c_{k}\), necessary for the construction of the dogleg step, is defined as  
\[c_{k}=-\bigg{(}\frac{\left\|\
abla f_{k}\right\|_{2}}{\left\|l_{k}\
abla f_{k }\right\|_{2}}\bigg{)}^{2}\
abla f_{k} \tag{18}\]  
and if \(c_{k}\geq\Delta_{k}\), a step \(s_{\text{trial}}\) is defined as  
\[s_{\text{trial}}=-\Delta_{k}\frac{\
abla f_{k}}{\left\|\
abla f_{k}\right\|_{ 2}}, \tag{19}\]  
Otherwise, \(s_{\text{trial}}\) is a convex combination of \(c_{k}\) and \(s_{k}^{N}\):  
\[s_{\text{trial}}=c_{k}+\tau\cdot(s_{k}^{N}-c_{k}), \tag{20}\]  
where \(\tau\) is the positive root of the 2 nd degree equation obtained, imposing that \(m_{k}(s_{k})=\Delta_{k}\),  
\[\|s_{k}^{N}-c_{k}\|^{2}\tau^{2}+(c_{k}^{T}\cdot s_{k}^{N}-||c_{k}||^{2})\tau+ ||c_{k}||^{2}-\Delta_{k}^{2}=0. \tag{21}\]  
The projected step \(s_{\text{trial,proj}}\) is  
\[s_{\text{trial,proj}}=\text{Proj}(x_{k}+s_{\text{trial}})-x_{k},\]  
where \(\text{Proj}(x)\) maps \(x\) into the box-bounded domain;  
\[\text{Proj}(x)=\max\{l,\min\{x,u\}\}. \tag{22}\]  
The projection map is employed at the very last step of the algorithm to achieve the best possible convergence ratio [18]. The first-order optimality conditions (5) are satisfied for each \(\{x_{k}\}\) if \(s_{k}\) meets the sufficient Cauchy decrease condition on the trust region model. \[\rho_{C}(s_{k})=\frac{m_{k}(0)-m_{k}(s_{k})}{m_{k}(0)-m_{k}\big{(}s_{k}^{C} \big{)}}\geq\varepsilon_{1},\quad\varepsilon_{1}\in(0,1). \tag{23}\]  
The trial step \(s_{k}\) is accepted if \(\rho_{C}(s_{\text{trial,proj}})\) meets condition (23). Otherwise, \(s_{k}\) is computed as a convex combination of \(s_{\text{trial,proj}}\) and \(s_{k}^{C}\):  
\[s_{k}=t\,s_{k}^{C}+(1-t)s_{\text{trial}},\quad t\in[0,1] \tag{24}\]  
with \(t\) chosen such that  
\[\rho_{C}(s_{k})=\rho_{c}\Big{(}t\,s_{k}^{C}+(1-t)s_{\text{trial,proj}}\Big{)} =\varepsilon_{1}. \tag{25}\]  
At the end of the trust-region subproblem, the predicted reduction of \(m_{k}\) and the actual reduction in \(f\) at \(x_{k}+s_{k}\) are checked using  
\[\rho_{f}(s_{k})=\frac{f(x_{k})-f(x_{k}+s_{k})}{m_{k}(0)-m_{k}(s_{k})}\geq \varepsilon_{2},\quad\varepsilon_{2}\in(0,1) \tag{26}\]If the trial point reduces the objective function, \(x_{k+1}=x_{k}+s_{k}\) is set, and the trust-region radius \(\Delta_{k+1}\) is updated, reducing it by a fixed fraction, ensuring it remains above a minimum value \(\Delta_{m1}\). If \(s_{k}\) does not satisfy (26), it is rejected and \(\Delta_{k}\) is reduced for improved model reliability.