Numerical methods using two different approximations of space-filling curves for black-box global optimization

In this paper, multi-dimensional global optimization problems are considered, where the objective function is supposed to be Lipschitz continuous, multiextremal, and without a known analytic expression. Two different approximations of Peano-Hilbert curve to reduce the problem to a univariate one satisfying the Hölder condition are dis- cussed. The first of them, piecewise-linear approximation, is broadly used in global optimization and not only whereas the second one, non- univalent approximation, is less known. Multi-dimensional geomet- ric algorithms employing these Peano curve approximations are intro- duced and their convergence conditions are established. Numerical experiments executed on 800 randomly generated test functions taken from the literature show a promising performance of algorithms em- ploying Peano curve approximations w.r.t. their direct competitors.