Convergence conditions and numerical comparison of global optimization methods based on dimensionality reduction schemes

This paper is devoted to numerical global optimization algorithms applying several ideas to reduce the problem dimension. Two approaches to the dimensionality reduction are considered. The first one is based on the nested optimization scheme that reduces the multidimensional problem to a family of one-dimensional subproblems connected in a recursive way. The second approach as a reduction scheme uses Peano-type space-filling curves mapping multidimensional domains onto one-dimensional intervals. In the frameworks of both the approaches, several univariate algorithms belonging to the characteristical class of optimization techniques are used for carrying out the one-dimensional optimization. Theoretical part of the paper contains a substantiation of global convergence for the considered methods. The efficiency of the compared global search methods is evaluated experimentally on the well-known GKLS test class generator used broadly for testing global optimization algorithms. Results for representative problem sets of different dimensions demonstrate a convincing advantage of the adaptive nested optimization scheme with respect to other tested methods.