## Special case of non-convex quadratically constrained quadratic programming

Sorry if this is not MO question, but had no luck at other places.

Consider a special case of QCQP problem:

$x^TPx + p^Tx \rightarrow \min_{x}$

s.t.

$x^TQ^{(k)}x = 0$ $k = 1...n$.

where $P$ is positive definite and $Q^{(k)}$ are matrices of the special kind, such that $x^TQ^{(1)}x = 0 \Leftrightarrow x_1x_4 - x_3x_2 = 0$, i.e. (I'm using matlab-like notation) $Q^{(1)} =$ [ [0 0 0 1/2]; [0 0 -1/2 0]; [0 -1/2 0 0]; [1/2 0 0 0]]

This problem appears in statistics/time series analysis/etc. Is there any hope that Lagrange SDP relaxation would be useful for solving this?

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