Adaptive Global Optimization Based on Nested Dimensionality Reduction

Abstract

In the present paper, the multidimensional multiextremal optimization problems and the numerical methods for solving these ones are considered. A general assumption only is made on the objective function that this one satisfies the Lipschitz condition with the Lipschitz constant not known a priori. The problems of this type are frequent in the applications. Two approaches to the dimensionality reduction for the multidimensional optimization problems were considered. The first one uses the Peano-type space-filling curves mapping a one-dimensional interval onto a multidimensional domain. The second one is based on the nested optimization scheme, which reduces a multi-dimensional problem to a family of the one-dimensional subproblems. A generalized scheme combining these two approaches has been proposed. In this novel scheme, solving a multidimensional problem is reduced to solving a family of problems of lower dimensionality, in which the space-filling curves are used. An adaptive algorithm, in which all arising subproblems are solved simultaneously has been implemented. The numerical experiments on several hundred test problems have been carried out confirming the efficiency of the proposed generalized scheme.