  One of the hard optimization problems that has a semi-definite relaxation
with quantitative bound on the approximation error is the maximization of a
convex quadratic form on the hypercube. The relaxation not only yields an upper
bound on the optimal value, but its solution can be used to construct random
sub-optimal solutions of the original problem whose expected value is not less
than $\frac{2}{\pi}$ times the value of the relaxation. This constant cannot be
improved globally. More precisely, for every $\epsilon > 0$ there exists a
problem instance for which the ratio of the two values in question is larger
than $\frac{\pi}{2} - \epsilon$. However, if a given problem instance is
considered, then the relaxation yields a concrete solution which may result in
a much better ratio. In this contribution we present an improved, explicit
bound depending on the rank of the solution. We consider also the problem of
maximization of a convex hermitian quadratic form on the complex poly-disc. In
this case a bound on the approximation error is given by the
$\frac{4}{\pi}$-theorem of Ben-Tal, Nemirovski, and Roos. The derivation of a
rank-dependent improved bound is similar to the real case. In the complex case
we provide explicit expressions in the form of an infinite series and
conjecture a closed-form expression.
