Mathematical Programming

, Volume 173, Issue 1–2, pp 251–280 | Cite as

Distributionally robust expectation inequalities for structured distributions

  • Bart P. G. Van ParysEmail author
  • Paul J. Goulart
  • Manfred Morari
Full Length Paper Series A


Quantifying the risk of unfortunate events occurring, despite limited distributional information, is a basic problem underlying many practical questions. Indeed, quantifying constraint violation probabilities in distributionally robust programming or judging the risk of financial positions can both be seen to involve risk quantification under distributional ambiguity. In this work we discuss worst-case probability and conditional value-at-risk problems, where the distributional information is limited to second-order moment information in conjunction with structural information such as unimodality and monotonicity of the distributions involved. We indicate how exact and tractable convex reformulations can be obtained using standard tools from Choquet and duality theory. We make our theoretical results concrete with a stock portfolio pricing problem and an insurance risk aggregation example.


Optimal inequalities Extreme distributions Convex optimisation Choquet representation CVaR 

Mathematics Subject Classification

90C34 90C15 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2017

Authors and Affiliations

  • Bart P. G. Van Parys
    • 1
    Email author
  • Paul J. Goulart
    • 2
  • Manfred Morari
    • 3
  1. 1.Operations Research CenterMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Department of Engineering ScienceUniversity of OxfordOxfordUK
  3. 3.School of Engineering and Applied ScienceUniversity of PennsylvaniaPhiladelphiaUSA

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