A Framework for Multiobjective and Uncertain Resource Allocation Problems in Survey Sampling based on Conic Optimization

In most textbooks optimal sample allocation is tailored to rather theoretical examples. However, in practice we often face large-scale surveys with conflicting objectives and many restrictions on the quality and cost at population and subpopulation levels. This multiobjectiveness results in a multitude of efficient sample allocations, each giving different weight to a single survey purpose. Additionally, since the input data to the allocation problem often relies on supplementary information derived from estimation, historical data, or expert knowledge, allocations might be inefficient when specified for sampling. This doctoral thesis presents a framework for optimal allocation to standard sampling schemes that allows for specifying the tradeoff between different objectives and analyzing their sensitivity to other problem components, aiming to support a decision-maker in identifying an at-most preferred sample allocation. It dedicates a full chapter to each of the following core questions: 1) How to efficiently incorporate quality and cost constraints for large-scale surveys, say, for thousands of strata with hundreds of precision and cost constraints? 2) How to handle vector-valued objectives with their components addressing different, possibly conflicting survey purposes? 3) How to consider uncertainty in the input data? The techniques presented can be used separately or in combination as a general problem-solving framework for constrained multivariate and multidomain, possibly uncertain, sample allocation. The main problem is formulated in a way that highlights the different components of optimal sample allocation and can be taken as a gateway to develop solution strategies to each of the questions above, while shifting the focus between different problem aspects. The first question is addressed through a conic quadratic reformulation, which can be efficiently solved for large problem instances using interior-point methods. Based on this the second question is tackled using a weighted Chebyshev minimization, which provides insight into the sensitivity of the problem and enables a stepwise procedure for considering nonlinear decision functionals. Lastly, uncertainty in the input data is addressed through regularization, chance constraints and robust problem formulations.