Let $P_1$, $P_2$ be two hermitian matrices. Can anyone comment about the following (QCQP) \begin{equation} \min_{z}~z^{H}z \\\ ~~subject~to ~z^{H}P_1z+1\leq 0,~z^{H}P_2z+1\leq 0 \end{equation} I am familiar with semi-definite relaxation. But I was wondering if we could do more here, since the objective is convex.

Let $P_1$, $P_2$ be two Hermitian matrices. Can anyone comment on the following QCQP?

$$\begin{array}{ll} \text{minimize} & z^{H} z\\ \text{subject to} & z^{H} P_1 z +1 \leq 0\\ & z^{H} P_2 z + 1 \leq 0\end{array}$$

I am familiar with semidefinite relaxation. But I was wondering if we could do more here, since the objective is convex.