Efficient global optimization of multivariate functions with unknown H\"{o}lder constants via \(α\)-dense curves

In this paper, we consider the global optimization problem of non-smooth functions over an \(n\)-dimensional box, satisfying the H\"{o}lder condition.We focus our study on the case when the H\"{o}lder constant is not a priori known. We develop and analyze two algorithms. The first one is an extended version of the Piyavskii's method that adaptively construct a linear secants of the parabolic support functions, avoiding the need for a known H\"{o}lder constant. The second algorithm employs a reducing transformation approach which consists of generating, in the feasible box, an α-dense curves, effectively converting the multivariate initial problem to a problem of a single variable. We prove the convergence of both algorithms. Their practical efficiency is evaluated through numerical experiments on some test functions and comparison with existing techniques.