Efficient and robust optimal design for quantile regression based on linear programming

When informing decisions with experimental data, it is often necessary to quantify the distribution tails of uncertain system responses using limited data. To maximize the information content of the data, one is naturally led to use experimental design. However, common design techniques minimize global statistics such as the average estimation or prediction variance. Novel methods for optimal experimental design that target distribution tails are developed. To achieve this, pre-asymptotic estimates of the data uncertainty are produced via an upper bound on a prescribed quantile, computed using quantile regression. Two optimal design problems are formulated: (i) Minimize the variance of the upper bound; and (ii) Minimize the Conditional Value-at-Risk of the upper bound. Additionally, each design problem is augmented with an added cardinality constraint to bound the number of experiments. These optimal design problems are reduced to continuous and mixed-integer linear programming problems. Consequently, the proposed methods are extremely efficient, even when applied to large datasets. The application of the proposed design formulation is demonstrated through a sensor placement problem in direct field acoustic testing.