## 1 Introduction  
There are plenty of real problems where the minimization of a twice continuously differentiable functional \(f:\mathbb{R}^{n}\rightarrow\mathbb{R}\) is sought, (possibly) subject to several linear and nonlinear constraints. Among authoritative textbooks, where such problems are widely detailed, we can surely find [1, 2, 3]. Such general problems typically require the solution to a sequence of simple sub-problems with the following pattern:  
\[ \begin{array}{ll}\min_{x}&\varphi_{k}(x)\\ \text{s.t.}&x\in D_{k},\end{array} \qquad\qquad D_{k}:\left\{ \begin{array}{ ll}A_{k}x+u_{k}=0\\ B_{k}x+v_{k}\leq 0,\end{array} \right. \tag{1}\]  
where \(A_{k}\in\mathbb{R}^{m_{k}\times n}\), \(B_{k}\in\mathbb{R}^{p_{k}\times n}\), \(u_{k}\in\mathbb{R}^{m_{k}}\), \(v_{k}\in\mathbb{R}^{p_{k}}\), \(m_{k},p_{k}\geq 1\) and \(k\geq 1\). Furthermore, \(\varphi_{k}(x)\) represents a _model_ of the smooth function \(f(x)\) at the current iterate \(x_{k}\) and the feasible set \(D_{k}\) represents a _linearization_ of the constraints. As is well known, affine and quadratic polynomials based on Taylor's expansion are often adopted to represent the models \(\{\varphi_{k}(x)\}\), but valid alternatives also include least squares approximations, Radial Basis Functions, metamodels based on Splines, B-Splines, Kriging, etc. [4, 5]. We remark that the advantage of solving the sequence of sub-problems (1) in place of the original nonlinearly constrained problem, within a suitable convergence framework, essentially relies on their simplicity. In particular, in this paper our focus is on investigating the role and the properties of the next problem (2), that represents a special sub-case of the more general problem (1). More specifically, we consider the case where in (1) the feasible set \(D_{k}\) includes only a _finite number of inequalities_, and the function \(\varphi_{k}(x)\) is a quadratic functional, i.e., we focus on the sub-problem and drop the dependency on \[ \begin{split}\min_{\alpha,\beta}&\varphi(x)=\frac{1}{2}x^{T}Qx+b^{T }x+c\\ x&=\tilde{x}+\alpha d+\beta z\\ a_{1}&\leq\alpha\leq b_{1}\\ a_{2}&\leq\beta\leq b_{2}\\ \epsilon_{1}\alpha+\epsilon_{2}\beta&\leq\epsilon_{3} \end{split}  \tag{2}\]  
where \(Q\in\mathbb{R}^{n\times n}\), \(b,\tilde{x}\in\mathbb{R}^{n}\), \(c\in\mathbb{R}\), \(d,z\in\mathbb{R}^{n}\) are given \(n\)-real search directions, and \(a_{1}\leq b_{1}\), \(a_{2}\leq b_{2}\). Despite the apparent specific structure of (2), a number of real applications may benefit from its solution, as partly described in Section 4 (see also [6] for a general perspective and [7] for a more recent similar viewpoint within neural network frameworks). As an example of versatility for the structure of (2), both in TRMs and LBMs, we will shortly consider how it may be possibly successfully embedded within the framework of [PERSON]'s methods (TNMs--see Table 1). Where (see also [8; 9; 10; 11; 12; 13; 14])  
* \(d\in\mathbb{R}^{n}\) represents an approximate Newton-type direction, at the current feasible point \(\tilde{x}\in\mathbb{R}^{n}\);
* \(z\in\mathbb{R}^{n}\) represents a negative curvature direction for the nonlinear function \(f(x)\), at the current feasible point \(\tilde{x}\in\mathbb{R}^{n}\);
* \(Q\in\mathbb{R}^{n\times n}\) represents the exact/approximate Hessian matrix of \(f(x)\) at \(\tilde{x}\);
* \(b\in\mathbb{R}^{n}\) represents the exact/approximate Gradient vector of \(f(x)\) at \(\tilde{x}\);
* \(\alpha\) and \(\beta\) are steplengths along the directions \(d,z\in\mathbb{R}^{n}\) (i.e., following the taxonomy of Table 1, we have \(\alpha\leftarrow\omega_{1}(\alpha)\) and \(\beta\leftarrow\omega_{2}(\alpha)\)), with \(-\infty<a_{1}\leq b_{1}<+\infty\) and \(-\infty<a_{2}\leq b_{2}<+\infty\).