Trust-Region (TR) algorithm for constrained nonlinear least squares problem

Step 1. Given initial iterative point $x_1\in R^m$ , $J\left(x_k\right)\in R^{n^2\times N_e}$ , ${\Delta }_1>0$ , $0<\epsilon <1$ , $\eta >0$ , $0<{\tau }_3<{\tau }_4<1<{\tau }_1$ , $0\le {\tau }_2<1$ and set $k:=1$ ;

Step 2. if ${\Vert J{\left(x_k\right)}^{\text{T}}G\left(x_k\right)\Vert }_2\le \epsilon$ then stop; otherwise, get $s_k$ from solving Equation (15);

Step3. Compute $r_k=\frac{Are{s}_k}{Pre{s}_k}$ ; $x_{k+1}=\{\begin{array}{l}{x}_k,\text{if}{r}_k<\eta ,\text{gotostep1}\\ x_k+s_k,\text{if}{r}_k\ge \eta ,\text{gotostep4}\end{array}$ ;

${\Delta }_{k+1}=\{\begin{array}{l}\left[{\tau }_3{\Vert s_k\Vert }_2,{\tau }_4{\Delta }_k\right],\text{if}{r}_k<{\tau }_2\\ \left[{\Delta }_k,{\tau }_1{\Delta }_k\right],\text{if}{r}_k\ge {\tau }_2\end{array}$ ;

Step 4. Compute a new function $\Psi \left(x_k+s_k\right)$ , $k:=k+1$ and go to step 2.