## 3 On the Use of Quadratic Sub-Problems Within TRMs and LBMs, in Large-Scale Optimization  
Here, we give details about a possible motivation for our proposal, in order to reduce the gap between two renowned classes of optimization methods, namely TRMs and LBMs. We are indeed persuaded that such a viewpoint may suggest a number of possible enhancements, to improve both the last classes of methods. In this regard, observe that a TRM for large-scale problems is an iterative procedure that generates the sequence of \(n\)-real iterates \(\{x_{k}\}\), and seeks at any step \(k\) for the solution of the trust-region sub-problem  
\[ \begin{array}{c}\min_{s}q_{k}(s)=f(x_{k})+\
abla f(x_{k})^{T}s+\frac{1}{2}s ^{T}Q_{k}s\\ \|s\|_{2}\leq\Delta_{k},\end{array}  \tag{4}\]  
where \(x_{k}\) is the current iterate, \(Q_{k}\) represents the _exact/approximate_ Hessian matrix \(\
abla^{2}f(x_{k})\), and \(\Delta_{k}>0\) represents the radius of the trust-region, i.e., the compact subset where the model \(q_{k}(s)\) needs to be validated (for an exhaustive description of TRMs for Nonlinear Programming, the reader can refer to [2]). A number of possible variants of (4) can be introduced when \(n\) is large, including iterative updating strategies for both \(Q_{k}\) and \(\Delta_{k}\), and a number of approximate/sophisticated/refined schemes for its solution are available in the literature. A distinguishing feature of TRMs, with respect to LBMs, is that at iteration \(k\) the methods in the first class attempt to determine the stepsize \(\alpha_{k}\) and the search direction \(d_{k}\) at once, so that \(x_{k+1}=x_{k}+s_{k}\equiv x_{k}+\alpha_{k}d_{k}\), where \(s_{k}\) indeed approximately/exactly solves (4). Conversely, in LBMs, the computations of \(\alpha_{k}\) and \(d_{k}\) are independent, as detailed later on in this paper. In particular, (see also [3]) the effective computation of \(s_{k}\) in TRMs properly attempts to comply with the following issues:  
* \(s_{k}\) can be computed by either an exact (small- and medium-scale problems) or an approximate (large-scale problems) procedure;
* In order to prove the global convergence of the sequence \(\{x_{k}\}\) to stationary limit points satisfying either first- or second-order necessary optimality conditions, \(s_{k}\) is  
Figure 1: A graphical representation of the feasible set in (2). The dashed-dotted lines represent all the extreme choices for the last inequality constraint in (2). required to provide a _sufficient reduction_ of the quadratic model \(q_{k}(s)\), i.e., the difference \(q_{k}(0)-q_{k}(s_{k})\) is asked to satisfy a condition like (\(c>0\)) \[q_{k}(0)-q_{k}(s_{k})\geq c\|\
abla f(x_{k})\|_{2}\min\biggl{\{}\Delta_{k},\frac {\|\
abla f(x_{k})\|_{2}}{1+\|Q\|_{2}}\biggr{\}};\] * \(s_{k}\) can be computed by an approximate procedure, e.g., by adopting a Cauchy step or using the Steihaug conjugate gradient (see [22; 23]), regardless of \(Q_{k}\) signature. Then, the approximate solution of (4) is merely sought on a _linear manifold_ of a dimension of one or at most two, rather than on the entire subset \(B\equiv\{s\in\mathbb{R}^{n}\;:\;\|s\|_{2}\leq\Delta_{k}\}\); * Depending on a number of additional assumptions, TRMs can prove to be globally convergent to either a simple stationary limit point, or to a point which satisfies second-order necessary optimality conditions [2]; * Finding the exact/accurate solution of the sub-problem (4) is in general quite a cumbersome task in large-scale problems, representing a difficult goal that is often (when possible) skipped. On the other hand, to some extent, LBMs represent the counterpart of TRMs. Indeed, to yield the next iterate \(x_{k+1}=x_{k}+\alpha_{k}d_{k}\), they perform the computation of the steplength \(\alpha_{k}\) and the direction \(d_{k}\) as separate tasks.