Feasibility of a given set of homogenuous nonconvex quadratic inequality constraints Let $C_1$,$C_2$,...$C_N$ be $M \times M$ indefinite hermitian matrices. What can we say about the following quadratic constriants
\begin{align}
w^{H}C_1w>0 \\\
w^{H}C_2w>0 \\\
...~~~~~~~~~~ \\\
...~~~~~~~~~~ \\\
w^{H}C_Nw>0 
\end{align}
where $w$ is a non-zero $M\times 1$ complex vector. Can we come up with conditions for existence of such a $w$? I tried using the CVX package to determine the same (after using semi-definite relaxation). It always gives the zero vector as the answer.
 A: As mentioned in its user's guide, CVX does not distinguish between constraints in the form of $f(x) < 0$ and $f(x) \leq 0 $, which is very common in numerical optimization. Instead, you should use 
$w^{H} C_i w \geq 1, \quad i = 1,2,\cdots,N$
to avoid the zero vector solution. Of course, you can replace $1$ by any positive constant. This is because we can always find $w$ via scaling so that the above conditions hold provided that the conditions in your original post are satisfied, i.e., there exists $\hat{w}$ such that 
$\hat{w}^{H} C_i \hat{w} = \epsilon_i > 0, \quad i = 1,2,\cdots,N$.
A: One option for a convex relaxation is to search for a positive semidefinite hermitian $W$ with $\mathrm{Trace}(WC_i)\geq 1$ for all $i$.  These conditions are equivalent to the ones you have written when $W$ is also rank one ($W = ww^H$), i.e., they are necessary but not sufficient for existence of the $w$ you seek.
If you like you could also try minimizing the nuclear norm of $W$ subject to these constraints because that is a heuristic convex surrogate for minimizing rank.  If the resulting optimal $W$ is not rank one you could also try for example taking the SVD of the optimal $W$ and see if any of the first few singular vectors has the property you seek.  
