I am facing the following problem: $$\begin{equation*} \begin{aligned} & \underset{z}{\text{minimize}} & & \textbf{z}^T \textbf{Q} \textbf{z} + \textbf{b}^T \textbf{z}\\ & \text{subject to} && \textbf{Az } \leq \textbf{d} \end{aligned} \end{equation*} $$
where $\textbf{Q}\geq 0$, this is, positive semidefinite. The dimension of vector $\textbf{z}$ is $3$ and the constraint matrix $\textbf{A}$ has $8$ rows.
The Lagrangian would look like:
$$L(\textbf{z},\lambda) = \textbf{z}^T \textbf{Q} \textbf{z} + \textbf{b}^T \textbf{z} + \lambda^T(\textbf{Az } - \textbf{d}) $$
Replacing with the corresponding values I would get:
$$L (x_1, x_2, x_3, \lambda) = x_1^2 +x_2^2 + (1-|h_1|)x_1 + (1-|h_2|)x_2 + \lambda_1(-x_1+x_3 - h_1) + \lambda_2(-x_1 -x_3 + h_1) + \lambda_3(-x_2+x_3 - h_2) + \lambda_4(-x_2-x_3 + h_2) - \lambda_5 x_1 + \lambda_6 x_1 - \lambda_7 x_2 + \lambda_8 x_2$$
I must note that $h_1$ and $h_2$ are constants.