I am trying to solve the following QCQP
$$\begin{array}{ll} \underset{x}{\text{minimize}} & x^T P_0 x + q_0^Tx + r_0\\ \text{subject to} & x^T P_1 x + r_1< 0\end{array}$$
where symmetric matrices $P_0, P_1$ are positive semi-definite and . I would like to know when this problem has a solution and in this case what the minimizer $x^*$ is.
On Wikipedia, I read that in my case the problem is convex and "can be readily solved using interior point methods, as done with semidefinite programming". The only problem is that my optimization courses go back a long way and I don't know how to write my problem in the form of a semidefinite program. Maybe there is a much more effective method, I don't know.
Any help would be most welcome. Thank you very much!