Algorithm 2.1. ANNATR (A new nonmonotone adaptive trust region method)

Step 0. Let $x_0\in R^n$ , a positive definite matrix $B_{\text{0}}\in R^{n\times n}$ , $\theta \in \left(\text{0,1}\right)$ , $\tilde{\Delta }>0$ , $h\in \left(0,1\right)$ , $\nu >0$ , $\lambda >1$ , ${\eta }_i\in \left(0,1\right)$ , $i=0,1,2,\cdots$ , $D_{\text{0}}=f\left(x_0\right)$ , a positive integer $\tilde{M}$ and $\delta >0$ . Set $k:=0$ .

Step 1. If $g_k\le \delta$ , then stop.

Step 2. Compute $q_k$ according to expression (1.6), $s_k$ by (1.7) and set $p=0$ .

Step 3. Compute ${\Delta }_k$ by (1.8), solve subproblem (1.2) to find the trial step $d_k$ and compute $r_k$ by (2.4).

Step 4. If $r_k<\upsilon$ , then $p=p+1$ . Set $x_{k+1}=x_k$ , $k:=k+1$ .Go to Step 3.

Step 5. Set $x_{k+1}=x_k+d_k$ .

Step 6. Update $B_k$ by BFGS update formula [42] .

Set $k:=k+1$ and go to Step 1.