Adaptive global optimization based on a block-recursive dimensionality reduction scheme

Multidimensional multiextremal optimization problems and numerical methods for solving them are studied. The objective function is supposed to satisfy the Lipschitz condition with an a priori unknown constant, which is the only general assumption imposed on it. Problems of this type often arise in applications. Two dimensionality reduction approaches to multidimensional optimization problems, i.e., the use of Peano curves (evolvents) and a recursive multistep scheme, are considered. A generalized scheme combining both approaches is proposed. In the new scheme, an original multidimensional problem is reduced to a family of lower-dimensional problems, which are solved using evolvents. An adaptive algorithm with the simultaneous solution of all resulting subproblems is implemented. Computational experiments on several hundred test problems are performed. In accordance with experimental evidence, the new dimensional reduction scheme is effective.