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


Quadratic optimization subject to fixed number of quadratic constraints is "easy", even without any convexity assumptions. The algorithms are polynomialtime, but in practice quite hard to implement efficiently. See e.g. this. The complex case can obviously be reduced to the real one. 

