## 3 Problem Definition
### The Mathematical Model  
The mathematical model comprises an objective function and a set of constraints:  
\[\min f(x)=\sum_{i=1}^{n}\sum_{j=1}^{n}\sigma_{ij}x_{i}x_{j} \tag{1}\]  
subject to:  
\[\sum_{i=1}^{n}r_{i}x_{i}\geq R \tag{2}\]  
\[\sum_{i=1}^{n}x_{i}=1 \tag{3}\]\[0\leq x_{i}\leq 1,\forall i\in\{1,2,\ldots,n\} \tag{4}\] \[k_{min}\leq\sum_{i=1}^{n}z_{i}\leq k_{max}\] (5) \[\varepsilon_{i}z_{i}\leq x_{i}\leq\delta_{i}z_{i},\forall i\in\{1,2, \ldots,n\}\] (6) \[0\leq\varepsilon_{i}\leq\delta_{i}\leq 1,\forall i\in\{1,2, \ldots,n\}\] (7) \[p_{i}\leq z_{i},\forall i\in\{1,2,...,n\}\] (8) \[z_{i}\leq Mx_{i},\forall i\in\{1,2,...,n\}\] (9) \[z_{i}\in\{0,1\},\forall i\in\{1,2,...,n\} \tag{10}\]  
Equation (1) describes the investor's objective function. The investor's objective is to minimize the portfolio variance. Equations (1)-(4) outline the basic (\"unconstrained\") optimization problem and determine the UEF. Specifically, Equation (2) provides the lower bound for the investor's required return. Equation (3) ensures that portfolio weights add up to unity. The purpose of Equation (4) is to regulate leveraged positions. This equation is justified on the grounds of the existing short selling regulations in a wide range of countries [65]. A number of realistic situations often require additional constraints that can be summarized by means of Equations (5)-(10). By solving the constrained optimization problem given by Equations (1)-(10) the so-called constrained efficient frontier (CEF) is obtained. Equation (5) formulates cardinality constraints. Equation (6) defines quantity constraints. The quantity of each asset \(a_{i}\) is confined to a given range. A minimum quantity of wealth invested in asset \(a_{i}\) is given by \(\varepsilon_{i}\). A maximum quantity of wealth invested in asset \(a_{i}\) is given by \(\delta_{i}\). Both parameters \(\varepsilon_{i}\) and \(\delta_{i}\) range from 0 to 1 (Equation (7)). Note that Equation (4) is redundant when Equations (6) and (7) are considered. Equations (8) and (9) imply that certain assets are pre-assigned in the portfolio. In particular, given a vector of \(n\) binary decision variables \(Z\) (Equation (10)) (where \(z_{i}\) takes on value 1 if included in the portfolio and 0 otherwise), and a binary vector \(P\) of pre-assignments (in which \(p_{i}\) takes on value 1 if pre-assigned and 0 otherwise), whenever asset \(a_{i}\) is pre-assigned, it has to be included in the portfolio (Equation (8)). In Equation (9), \(M\) is a large positive value such that \(Mx_{i}\geq 1\) for all \(x_{i}>0\). Thus, if the quantity in the portfolio of asset \(a_{i}\), \(x_{i}\), is equal to 0, it means that this asset is not included in the portfolio (i.e., \(z_{i}=0\)). Only a few papers have considered the use of pre-assignment constraints. An example is [41], who found that the CEF obtained by pre-assigning a high-yield asset strictly dominates the CEF obtained by pre-assigning a low-yield asset. In general, they find an inverse relation between the average percentage loss (with regard to the UEF) and the return value. However, they do not report computing times for the CEF with pre-assignment constraints, which makes impossible to complete a fair comparison with their results.