Lipschitz and Hِlder global optimization using space-filling curves

In this paper, the global optimization problem min y ∈ S F ( y ) with S = [ a , b ] , a , b ∈ R N , and F ( y ) satisfying the Lipschitz condition, is considered. To deal with it four algorithms are proposed. All of them use numerical approximations of space-filling curves to reduce the original Lipschitz multi-dimensional problem to a univariate one satisfying the Hölder condition. The Lipschitz constant is adaptively estimated by the introduced methods during the search. Local tuning on the behavior of the objective function and a newly proposed technique, named local improvement, are used in order to accelerate the search. Convergence conditions are given. A theoretical relation between the order of a Hilbert space-filling curve approximation used to reduce the problem dimension and the accuracy of the resulting solution is established, as well. Numerical experiments carried out on several hundreds of test functions show a quite promising performance of the new algorithms.