Global minimization algorithms for Holder functions

This paper deals with the one-dimensional global optimization problem where the objective function satisfies H\"{o}lder condition over a closed interval. A direct extension of the popular Piyavskii method proposed for Lipschitz functions to H\"{o}lder optimization requires an a priori estimate of the H\"{o}lder constant and solution to an equation of degree $N$ at each iteration. In this paper a new scheme is introduced. Three algorithms are proposed for solving one-dimensional H\"{o}lder global optimization problems. All of them work without solving equations of degree $N$. The case (very often arising in applications) when H\"{o}lder constant is not given a priori is considered. It is shown that local information about the objective function used inside the global procedure can accelerate the search significantly. Numerical experiments show quite promising performance of the new algorithms.