Discrete Approximation and Quantification in Distributionally Robust Optimization

Yongchao Liu (), Alois Pichler and Huifu Xu ()
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Yongchao Liu: School of Mathematical Sciences, Dalian University of Technology, 116024 Dalian, China
Alois Pichler: Faculty of Mathematics, Chemnitz University of Technology, 09126 Chemnitz, Germany
Huifu Xu: School of Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, United Kingdom

Mathematics of Operations Research, 2019, vol. 44, issue 1, 19-37

Abstract: Discrete approximation of probability distributions is an important topic in stochastic programming. In this paper, we extend the research on this topic to distributionally robust optimization (DRO), where discretization is driven by either limited availability of empirical data (samples) or a computational need for improving numerical tractability. We start with a one-stage DRO where the ambiguity set is defined by generalized prior moment conditions and quantify the discrepancy between the discretized ambiguity set and the original one by employing the Kantorovich/Wasserstein metric. The quantification is achieved by establishing a new form of Hoffman’s lemma for moment problems under a general class of metrics—namely, ζ -structures. We then investigate how the discrepancy propagates to the optimal value in one-stage DRO and discuss further the multistage DRO under nested distance. The technical results lay down a theoretical foundation for various discrete approximation schemes to be applied to solve one-stage and multistage distributionally robust optimization problems.

Keywords: Hoffman’s lemma; Kantorovich/Wasserstein metric; discretization of ambiguity set; moment conditions; nested distance (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (5)

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