  This paper develops new semidefinite programming (SDP) relaxation techniques
for two classes of mixed binary quadratically constrained quadratic programs
(MBQCQP) and analyzes their approximation performance. The first class of
problem finds two minimum norm vectors in $N$-dimensional real or complex
Euclidean space, such that $M$ out of $2M$ concave quadratic functions are
satisfied. By employing a special randomized rounding procedure, we show that
the ratio between the norm of the optimal solution of this model and its SDP
relaxation is upper bounded by $\frac{54M^2}{\pi}$ in the real case and by
$\frac{24M}{\sqrt{\pi}}$ in the complex case. The second class of problem finds
a series of minimum norm vectors subject to a set of quadratic constraints and
a cardinality constraint with both binary and continuous variables. We show
that in this case the approximation ratio is also bounded and independent of
problem dimension for both the real and the complex cases.
