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SPEPS Preprint

Author(s): Alois Pichler, TU Chemnitz
Huifu Xu, University of Southampton
Title: Quantitative Stability Analysis for Minimax Distributionally Robust Risk Optimization
Date of Acceptance: 19.04.2017
Submission Date: 12.04.2017
Series Title: Stochastic Programming E-Print Series
(SPEPS)
Editors: Julie L. Higle; Werner Römisch; Surrajeet Sen
Complete Preprint: pdf (urn:nbn:de:kobv:11-100246239)
Keywords (eng): distortion risk measure, &zeta-ball, Wasserstein ball, quantitative stability analysis
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Abstract (eng):
This paper considers distributionally robust formulations of a two stage stochastic programming problem with the objective of minimizing a distortion risk of the minimal cost incurred at the second stage.We carry out stability analysis by looking into variations of the ambiguity set under theWasserstein metric, decision spaces at both stages and the support set of the random variables. In the case when it is risk neutral, the stability result is presented with the variation of the ambiguity set being measured by generic metrics of ζ-structure, which provides a unified framework for quantitative stability analysis under various metrics including total variation metric and Kantorovich metric. When the ambiguity set is structured by a ζ-ball, we find that the Hausdorff distance between two ζ-balls is bounded by the distance of their centres and difference of their radius. The findings allow us to strengthen some recent convergence results on distributionally robust optimization where the centre of the Wasserstein ball is constructed by the empirical probability distribution.
 
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