A two-phase approach in a global optimization algorithm using multiple estimates of Hölder constants

In this paper, the global minimization problem of a multi-dimensional black-box Lipschitzian function is considered. In order to pass from the original Lipschitz multi-dimensional problem to a univariate one, an approach using space-filling curves to reduce the dimension is applied. The method does not use derivatives and, at each iteration, works with a set of estimates of the Hölder constant of the reduced one-dimensional problem. A two-phase technique is applied to accelerate the search of the global minimum. Numerical experiments carried out on several hundreds of test functions show a promising performance of the discussed algorithm in comparison with its direct competitors.