I am struggling with the following optimization problem:
\begin{equation*} \begin{aligned} & \underset{x_{1}, \ ..., \ x_{5} \in \mathrm{R}}{\text{minimize}} & & \begin{pmatrix} x_{1} + x_{2} \\ x_{1} x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \end{pmatrix}^{\textbf{T}} \textbf{A} \begin{pmatrix} x_{1} + x_{2} \\ x_{1} x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \end{pmatrix} + \textbf{B}^{\textbf{T}} \begin{pmatrix} x_{1} + x_{2} \\ x_{1} x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \end{pmatrix} \\ & \text{subject to} & & x_{1} \geq k \text{ and } x_{2} \geq k \end{aligned} \end{equation*} the matrix A is invertible.
My attemp: I introduce two artificial decision variables $x_{1}' = x_{1} + x_{2}$ and $x_{2}' = x_{1}x_{2}$ and I solve the quadratic programming problem:
\begin{equation*} \begin{aligned} & \underset{x_{1}', \ x_{2}', \ x_{3}, \ ..., \ x_{5} \in \mathrm{R}}{\text{minimize}} & & \begin{pmatrix} x_{1}'\\ x_{2}' \\ x_{3} \\ x_{4} \\ x_{5} \end{pmatrix}^{\textbf{T}} \textbf{A} \begin{pmatrix} x_{1}' \\ x_{2}' \\ x_{3} \\ x_{4} \\ x_{5} \end{pmatrix} + \textbf{B}^{\textbf{T}} \begin{pmatrix} x_{1}' \\ x_{2}' \\ x_{3} \\ x_{4} \\ x_{5} \end{pmatrix} \\ & \text{subject to} & & x_{1}' \geq 2k \text{ and } x_{2}' \geq k^{2} \end{aligned} \end{equation*}
Unfortunatlly, it produces wrong results since
\begin{equation*}
\left\{
\begin{array}{r c l}
x_{1}' & \geq & 2k\\
x_{2}' & \geq & k^{2}
\end{array}
\right.
\text{ does not imply }
\left\{
\begin{array}{r c l}
x_{1} & \geq & k\\
x_{2} & \geq & k
\end{array}
\right.
\end{equation*}
So, is it possible to overcome this issue? or am I investigating the wrong way?