Constructing Uncertainty Sets for Robust Linear Optimization

In this paper, we propose a methodology for constructing uncertainty sets within the framework of robust optimization for linear optimization problems with uncertain parameters. Our approach relies on decision-maker risk preferences. Specifically, we utilize the theory of coherent risk measures initiated by Artzner et al. [3], and show that such risk measures, in conjunction with the support of the uncertain parameters, are equivalent to explicit uncertainty sets for robust optimization. We also explore the structure of these sets. In particular, we study a class of coherent risk measures, called spectral risk measures, which give rise to polyhedral uncertainty sets of a very specific structure which is tractable in the context of robust optimization. In the case of discrete distributions with rational probabilities, which is useful in practical settings when we are sampling from data, we show that the class of all spectral risk measures (and their corresponding polyhedral sets) are finitely generated. A subclass of the spectral measures corresponds to polyhedral uncertainty sets symmetric through the sample mean. We show that this subclass is also finitely generated and can be used to find inner approximations to arbitrary, polyhedral uncertainty sets.