A new extension of Piyavskii's method to Hölder functions of several variables

In this paper we suggest a new extension of the method of Piyavskii for global optimization of a Hölder function with exponent 1β(β>1). In the one-dimensional case a modification of Piyavskii's algorithm is introduced and it is based on the construction of sub-estimators which are piecewise linear. The algorithm is then some what easily. Moreover, the results we obtain seem interesting. In the higher dimension, a new variant of the Alienor reducing transformation, which has been devised for exploiting one-dimensional global optimization techniques known for their great efficiency, is used. The method consists in reducing a multidimensional problem to a one-dimensional one using the so called α-dense curves. The convergence of the methods is also studied.