Title: Numerical optimization for multivariate optimal allocation problems with several levels of strata
Authors: Martin Rupp - University of Trier (Germany) [presenting]
Ralf Muennich - University of Trier (Germany)
Abstract: The aim of modern surveys is to provide accurate information on a large variety of variables as well as on different regional levels and other subclasses of the population. Hence, optimizing a stratified sampling design, optimal allocation of the sample size has to consider a vast number of strata along with optimization conflicts due to the complementary information of the variables of interest with regard to different levels of strata. Furthermore, particular quality or cost restrictions might be taken into account. Modelling this multivariate optimal allocation problem leads to a high dimensional multi-objective optimization problem with equality constraints and box-constraints for the stratum specific sample sizes. Taking advantage of the special structure of the variance functions and applying Pareto-optimization, the problem can be equivalently reformulated as a significantly lower dimensional non-linear system of equations, depending only on the Lagrange multipliers. Even though this system is non-smooth, it can be solved applying a semi-smooth newton algorithm with appropriate starting point and step size strategies. Due to the lower dimension, computational time is reduced considerably. The performance of the developed algorithm is tested on a business data set.