Space-filling curves and multiple estimates of Hölder constants in derivative-free global optimization

In this paper the global optimization problem where the objective function is multiextremal and satisfying the Lipschitz condition over a hyperinterval is considered. An algorithm that uses Peano-type space-filling curves to reduce the original Lipschitz multi-dimensional problem to a univariate one satisfying the Hölder condition is proposed. The algorithm at each iteration applies a new geometric technique working with a number of possible Hölder constants chosen from a set of values varying from zero to infinity showing so that ideas introduced in a popular DIRECT method can be used in the Hölder global optimization, as well. Convergence condition are given. Numerical experiments show quite a promising performance of the new technique.