Improving exploration strategies in large dimensions and rate of convergence of global random search algorithms

We consider global optimization problems, where the feasible region $\text{X}$ is a compact subset of ${\mathbb{R}}^d$ with $d\ge 10$. 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~$\text{X}$ is much less efficient than uniform sampling on a suitable subset of $\text{X}$, and (b) the effect of replacement of random points by low-discrepancy sequences is negligible.