Robust optimization with order statistic uncertainty set

Abstract

In this paper, we propose a new uncertainty set for robust models of linear optimization problems. We first study data-free and distribution-free statistical properties of continuous and independent random variables using the Probability Integral Transform. Based on these properties, we construct a new uncertainty set by placing constraints on the order statistics of random variables. We utilize the quantiles of random variables to depict the uncertainties and then adopt the formulation of the assignment problem to develop a tractable formulation for the order statistic uncertainty set. We show that the robust optimization models with the interval uncertainty set, the budget uncertainty set, and the demand uncertainty set can be obtained as special cases of the robust optimization model with the order statistic uncertainty set. Finally, using a robust portfolio construction problem as an example, we show via numerical experiments that the order statistic uncertainty set has better performance than other uncertainty sets when the sample size is small and the correlation between random variables is low.