## 1 Introduction
The constraint \(\epsilon_{1}\alpha+\epsilon_{2}\beta\leq\epsilon_{3}\) potentially plays a multi-purpose role, modeling for instance the _gradient-related_ property for the search direction \(\alpha d+\beta z\in\mathbb{R}^{n}\) at \(\tilde{x}\), i.e., \[(\alpha d+\beta z)^{T}\
abla f(\tilde{x})\leq-c\|\
abla f(\tilde{x})\|^{h},\] (3) \begin{table}
\begin{tabular}{l} \hline Set \(x_{0}\in\mathbb{R}^{n}\) \\ Set \(\eta_{k}\in[0,1)\) for any \(k\), with \(\{\eta_{k}\}\to 0\) \\
**OUTERITATIONS** \\
**for**\(k=0,1,\ldots\) \\   Compute \(b\approx\
abla f(x_{k})\) and \(Q\approx\
abla^{2}f(x_{k})\); if \(\|b\|\) is small then STOP \\
**INNER ITERATIONS** \\  - Compute \(d_{k}\), which approximately solves [PERSON]’s equation \\  \(Qd+b=0\), i.e., it satisfies the _truncation rule_\(\|Qd_{k}+b\|\leq\eta_{k}\|b\|\) \\  - Possibly compute a bounded negative curvature direction \(z_{k}\) at \(x_{k}\) \\ \end{tabular}
\end{table} 
Table 1: A standard framework for linesearch-based TNMs for large-scale problems. The alternative of possibly using negative curvature directions allows for convergence to stationary limit points which fulfill second-order necessary optimality conditions. being  
\[\left\{ \begin{array}{l}\varepsilon_{1}=d^{T}\
abla f(\tilde{x}),\\ \\ \varepsilon_{2}=z^{T}\
abla f(\tilde{x}),\\ \\ \varepsilon_{3}=-\tilde{c}\|\
abla f(\tilde{x})\|^{h},\qquad\tilde{c},h>0.\end{array} \right.\]  
The availability of an (exact) global solution for (2) may also suggest some alternatives to Table 1, either selecting a TRM or a LBM framework, or combining the two approaches. In particular, the scheme in Table 2 represents an immediate _acceleration scheme_ for linesearch-based TNMs with respect to Table 1, in case the global convergence of \(\{x_{k}\}\) to stationary limit points is simply sought. Note that selecting negative values for \(a_{1},a_{2}\) and positive ones for \(b_{1},b_{2}\) allows us to possibly perform the following:  
* reverse the directions \(d_{k}\) and \(z_{k}\);
* use (2) in the light of simulating a _dogleg-like_[3] procedure for TRMs, also in LBM nonconvex frameworks. In Table 2, the global convergence to stationary limit points is easily preserved by using similar results adopted for Table 1.  
As a further alternative case for considering the exact global solution of (2), with respect to Tables 1 and 2, we have the scheme in Table 3, where we suitably combine the strategies used in TRMs and LBMs to ensure global convergence (in particular, TRMs require the fulfillment of a sufficient reduction of the model in order to force a sufficient decrease in the objective function, so that they do not need any linesearch procedure, possibly implying a reduced computational burden with respect to LBMs. Conversely, LBMs easily compute an effective search direction but they need to perform a linesearch procedure, because they do not include any -direct- function reduction mechanism based on the local quadratic model). In particular, if the test \(\mathit{Ared}_{k}/\mathit{Pred}_{k}>\rho\) is fulfilled, there is no need to perform a linesearch procedure, since the global convergence for \(\{x_{k}\}\) is preserved by the trust-region framework. We also remark that, in Table 3, the computation of both \(\varphi(x_{k})\) and \(\varphi(x_{k}+v_{k})\) is required, regardless of the outcomes of the test \(\mathit{Ared}_{k}/\mathit{Pred}_{k}\
ot>\rho\), since in any case these quantities must be computed.