## 3 On the Use of Quadratic Sub-Problems Within TRMs and LBMs, in Large-Scale Optimization
Furthermore, unlike for TRMs, the novel iterate \(x_{k+1}\) in LBMs can be also obtained by adopting the more general update \[x_{k+1}=x_{k}+\alpha_{k}d_{k}+\beta_{k}z_{k},\] (5) with \(d_{k}\) and \(z_{k}\) now being two search directions summarizing different information on the function \(f(x)\), and \(\alpha_{k}\) and \(\beta_{k}\) being stepsizes. In particular: * when \(z_{k}\equiv 0\) (or \(\beta_{k}\equiv 0\) for any \(k\)), then \(d_{k}\) represents a Newton-type direction, being typically computed by approximately solving [PERSON]'s equation \(\
abla^{2}f(x_{k})d=-\
abla f(x_{k})\) at the current iterate \(x_{k}\). Then, an Armijo-type linesearch procedure is applied along \(d_{k}\) to compute \(\alpha_{k}\), provided that \(d_{k}\) is gradient-related (see e.g., [3]) at \(x_{k}\); * when \(z_{k}\
eq 0\), then \(d_{k}\) represents a Newton-type direction again, while \(z_{k}\) is typically a _negative curvature direction_ for \(f(x)\) at \(x_{k}\), which approximates an eigenvector associated with the least negative eigenvalue of \(\
abla^{2}f(x_{k})\). The vector \(z_{k}\) plays an essential role, when LBMs' convergence to stationary points satisfying the second-order necessary optimality conditions needs to be proved. In the last case, the computation of the steplengths \(\alpha_{k}\) and \(\beta_{k}\) is often carried out at once, (as in _curvilinear_ linesearch procedures--see [24]), or the steplength computation is carried out by pursuing independent tasks (see, for example, [25]). We highlight that in (5), when both \(d_{k}\
eq 0\) and \(z_{k}\
eq 0\), we may experience difficulties related to properly scaling the two search directions. As a general class of efficient algorithms within LBMs for large-scale problems, we find Truncated Newton methods (TNNs) coupled with a linesearch procedure (see Table 1). Similarly to general TRMs, they are evidently based on possibly computing \(d_{k}\) and \(z_{k}\) after exploiting the second-order Taylor's expansion of \(f(x)\) at \(x_{k}\). However, a couple of quite disappointing issues arise when applying linesearch-based TNOMs, namely: * Unlike trust-region based TNOMs, at iterate \(x_{k}\), the search of a stationary point for a quadratic polynomial model of \(f(x)\) (i.e., [PERSON]'s equation) is performed on \(\mathbb{R}^{n}\), so that the quadratic expansion is not trusted on a more reliable compact subset (trust-region) of \(\mathbb{R}^{n}\). Thus, the search direction \(d_{k}\) might show poor performance when the iterates in the sequence \(\{x_{k}\}\) are far from a stationary limit point \(x^{*}\). More specifically, note that in case \(\
abla^{2}f(x_{k})\succeq 0\); then, solving [PERSON]'s equation and the trust-region sub-problem \[\min_{d}\;\;q_{k}(d)=f(x_{k})+\
abla f(x_{k})^{T}d+\frac{1}{2}d^{T}\
abla^{2}f (x_{k})d\] \[\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\  
for any \(\gamma_{k}\geq\|[\
abla^{2}f(x_{k})]^{+}\
abla f(x_{k})\|_{2}\) yields the same solutions. Conversely, when \(\
abla^{2}f(x_{k})\) is indefinite, then [PERSON]'s equation provides a saddle point for \(q_{k}(d)\), that might be interpreted as a solution to a trust-region sub-problem (the interested reader may consider the paper [26] for some extensions). Furthermore, from this perspective, we remark that in LBMs, solving (2) where \(d_{k}=-\
abla f(x_{k})\), \(z_{k}\equiv 0\), and \(\varepsilon_{1}=\varepsilon_{2}=\varepsilon_{3}=0\), is to a to large extent equivalent to computing the Cauchy step when solving (4). Indeed, in the last case, the trust-region constraint in (4), in principle, can be equivalently replaced by the compact feasible set (box constraints) in (2), after setting \(\varepsilon_{1}=\varepsilon_{2}=\varepsilon_{3}=0\).