  We consider global optimization problems, where the feasible region $\X$ is a
compact subset of $\mathbb{R}^d$ with $d \geq 10$. For these problems, we
demonstrate the following. First: the actual convergence of global random
search algorithms is much slower than that given by the classical estimates,
based on the asymptotic properties of random points. Second: the usually
recommended space exploration schemes are inefficient in the non-asymptotic
regime. Specifically, (a) uniform sampling on entire~$\X$ is much less
efficient than uniform sampling on a suitable subset of $\X$, and (b) the
effect of replacement of random points by low-discrepancy sequences is
negligible.
