Consider the following quadratic program
$$\begin{array}{ll} \underset{x_1,x_2,x_3}{\text{minimize}} & x_1 x_2 +\frac{1}{2} x_1^2 + \frac32 x_2^2 + 2 x_3^2 + 2 x_1 + x_2 + 3x_3\\ \text{subject to} & x_1 + x_2 + x_3 = 1\\ & x_1 - x_2=0\\ & x_1, x_2, x_3 \geq 0\end{array}$$
How can I write the objective function in a form of $$\min_x \frac{1}{2}x^TQx+C^Tx$$
Second, we know that when $Q$ is a positive semidefinite matrix, i.e., when $y^TQy\geq 0$ for all $y$, the objective function is a convex function of $x$. Since the feasible set is a polyhedral set, it is a convex set. How can I show this for this specific problem?
Thank you for your help in advance!