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.



A certain type of quadratic constrained quadratic program (QCQP)
Yes, a lot can be said for your special case. Given the notation, I presume you are optimizing over $\mathbb{C}^n$. In this case, see Section 2 in the paper [Strong Duality in Nonconvex Quadratic Optimization with two Quadratic Constraints](http://epubs.siam.org/doi/abs/10.1137/050644471), A. Beck and Y. Eldar, *SIAM J. Optimization*, 17(3), 2006.


PS: The complex variable case seems to be easier than the real variable case.


