## 1 Introduction
\begin{table}
\begin{tabular}{l} \hline Set \(x_{0}\in\mathbb{R}^{n}\) \\ Set \(\eta_{k}\in[0,1)\) for any \(k\), with \(\{\eta_{k}\}\to 0\) \\
**OUTER ITERATIONS** \\
**for**\(k=0,1,\dots\) \\   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_\(\|Qd_{k}+b\|\leq\eta_{k}\|b\|\) \\  -  Set \(z_{k}=-b\) \\   Compute \(\alpha^{*}\) and \(\beta^{*}\) by solving (2); then, update the trust-region \\   parameters \(a_{1},a_{2},b_{1},b_{2}\) \\   Set \(v_{k}=\alpha^{*}d_{k}+\beta^{*}z_{k}\), and use an _Armijo-type procedure_ to select the \\   steplength \(\alpha_{k}\) along the direction \(v_{k}\) \\   Update \(x_{k+1}=x_{k}+\alpha_{k}v_{k}\) \\
**endfor** \\ \hline \end{tabular}
\end{table} 
Table 2: A standard framework for linesearch-based TNMs for large-scale problems which exploits the sub-problem (2). Differences with respect to Table 1 are quite evident. Finally, there is a chance to further exploit the scheme (2) in a TNM framework based on the linesearch procedure, in order to ensure global convergence properties for the sequence \(\{x_{k}\}\) to stationary limit points satisfying the _second-order necessary optimality conditions_ (namely, those stationary points where the Hessian matrix is positive and semidefinite). The resulting scheme is proposed in Table 4 and potentially does not require additional comments. The above examples give an overview of the possible basic contexts where the solution of the sub-problem (2) is sought. Hence, to some extent, specifically exploiting issues on its solution may yield a tool for practitioners working in Nonlinear Programming frameworks. We remark that, both in Sections 3 and 6, the reader may find additional guidelines for possible alternatives and extensions to the use of global solutions of (2). \begin{table}
\begin{tabular}{l} \hline Set \(x_{0}\in\mathbb{R}^{n}\) \\ Set \(\eta_{k}\in[0,1)\) for any \(k\), with \(\{\eta_{k}\}\to 0\) \\
**OUTER ITERATIONS** \\
**for**\(k=0,1,\dots\) \\   Compute \(b\approx\
abla f(x_{k})\) and \(Q\approx\
abla^{2}f(x_{k})\); if \(\|b\|\) is small then STOP \\
**INNER ITERATIONS** \\   Compute \(b\approx\
abla f(x_{k})\) and \(Q\approx\
abla^{2}f(x_{k})\); if \(\|b\|\) is small then STOP \\
**INNER ITERATIONS** \\   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_\(\|Qd_{k}+b\|\leq\eta_{k}\|b\|\) \\  - Set \(z_{k}=-b\) \\   Compute \(\alpha^{*}\) and \(\beta^{*}\) by solving (2); then, set \(v_{k}=\alpha^{*}d_{k}+\beta^{*}z_{k}\). Update the \\  \(Ared_{k}=f(x_{k})-f(x_{k}+v_{k})\), \(Pred_{k}=\varphi(x_{k})-\varphi(x_{k}+v_{k})\) \\   If \(Ared_{k}/Pred_{k}\
ot>\rho\), use an _Armijo-type procedure_ to select the steplength \\  \(\alpha_{k}\) along \(v_{k}\); otherwise, skip the linesearch procedure \\   Update the trust-region parameters \(a_{1},a_{2},b_{1},b_{2}\) \\   Update \(x_{k+1}=x_{k}+\alpha_{k}v_{k}\) \\
**endfor** \\ \hline \end{tabular}
\end{table} 
Table 3: A framework for combining trust-region and linesearch approaches within TNMs for large-scale problems, exploiting the sub-problem again (2). Differences with respect to Tables 1 and 2 are quite evident. We also highlight that our perspective both differs from SQP (Sequential Quadratic Programming) methods--see, for example, the seminal paper [15], and approaches from the literature where LSMs and TRMs have been combined. Indeed, in the basic structure of SQPs (see also [16; 17]), inner and outer iterations are performed. At each outer iteration, the pair given by primal-dual variables is computed, and a problem similar to (2) is addressed.