## 3 On the Use of Quadratic Sub-Problems Within TRMs and LBMs, in Large-Scale Optimization
On the other hand, in case \(\
abla^{2}f(x_{k})\succeq 0\), and setting (2) \(d_{k}=-\
abla f(x_{k})\) and \(z_{k}=-[\
abla^{2}f(x_{k})]^{+}\
abla f(x_{k})\), along with \(\varepsilon_{1}=\varepsilon_{2}=\varepsilon_{3}=0\), then, with similar reasoning, the solution to (2) closely resembles the application of the _dogleg method_ when solving (4). Finally, since the coefficients \(a_{1},a_{2}\) in (2) may have negative values, we may potentially reverse the directions \(d_{k}\) and \(z_{k}\) when solving (2). Thus, following the idea behind (3), the scheme (2) suggests that, in case \(\
abla^{2}f(x_{k})\) is also indefinite, (2) easily generalizes the proposals in [27]. In fact, following (3), we are able to exactly compute a global minimum \((\alpha^{*},\beta^{*})\) for (2), regardless of the signature of \(Q\), so that the resulting direction \(\alpha^{*}d_{k}+\beta^{*}z_{k}\) is gradient-related at \(x_{k}\). * As in (5), the search directions \(d_{k}\) and \(z_{k}\) might be suitably combined in a curvilinear framework (see, for example, [24]). However, to our knowledge, the selection of \(\alpha_{k}\) and \(\beta_{k}\) in the literature is seldom performed with a joint procedure to separately assess \(\alpha_{k}\) and \(\beta_{k}\), i.e., \(\alpha_{k}\) and \(\beta_{k}\) are rarely chosen as independent parameters. Hence, in the literature of linesearch-based TNOMs, the linesearch procedure that starts from \(x_{k}\) and yields \(x_{k+1}\) explores a one-dimensional manifold (regular curve), rather than considering \(x_{k}+\alpha d_{k}+\beta z_{k}\) as a two-dimensional manifold with independent real coefficients \(\alpha\) and \(\beta\). In this regard, using (2) within LBMs tends to partially compensate for the drawbacks in the last two items, in light of the great success that TRMs have achieved in the last decade. In particular, using (2) within linesearch-based TNOMs, our aim is that of developing a simple tool which could possibly carry out the following:  
1. Combines at iterate \(x_{k}\) two independently computed vectors, namely \(d_{k},z_{k}\in\mathbb{R}^{n}\), by exactly computing a _global minimum_ (we recall that, conversely, a global solution of the trust-region sub-problem (4) is often only approximately computed) for the two-dimensional constrained problem (2), being \(\bar{x}\gets x_{k}\), \(d\gets d_{k}\), \(z\gets z_{k}\);
2. Adaptively updates the parameters \(a_{1}\), \(a_{2}\), \(b_{1}\), \(b_{2}\) in (2), when the iterate \(x_{k}\) changes, following the rationale behind the update of \(\Delta_{k}\) in (4), and retaining the strong convergence properties of TRMs. This fact is of remarkable interest, since in (2) the information associated with the search directions \(d_{k}\) and \(z_{k}\) is suitably trusted in a _compact subset_ of \(\mathbb{R}^{n}\) (namely, the box constraints \(a_{1}\leq\alpha\leq b_{1}\), \(a_{2}\leq\beta\leq b_{2}\));
3. _Exactly computes a cheap global minimum_\((\alpha^{*},\beta^{*})\) for (2), so that the vector \(\alpha^{*}d_{k}+\beta^{*}s_{k}\) is then provided to a standard linesearch procedure such as the Armijo rule, to ensure that the global convergence of the sequence \(\{x_{k}\}\) to stationary (limit) points is preserved;
4. Allows for the convergence of subsequences of the iterates \(\{x_{k}\}\) to stationary limit points, where either first- or second-order necessary optimality conditions are fulfilled;
5. _Preserves generality_ within a wide range of optimization frameworks, as reported in the next Section 4;
6. _Combines_ the effects of \(d_{k}\) and \(z_{k}\), skipping all the drawbacks related to a possible different scaling between these directions. We recall that since \(d_{k}\) and \(z_{k}\) are generated through the application of different methods, then the comparison of their performances may be biased by the latter generating methods. The TNOMs sketched in Tables 2-4 are only examples of proposals in light of the previous comments.