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

Improving the constant in Nesterov's $\frac{\pi}{2}$-theorem