Mathematical Programming

, Volume 173, Issue 1–2, pp 151–192 | Cite as

Ambiguous risk constraints with moment and unimodality information

  • Bowen Li
  • Ruiwei JiangEmail author
  • Johanna L. Mathieu
Full Length Paper Series A


Optimization problems face random constraint violations when uncertainty arises in constraint parameters. Effective ways of controlling such violations include risk constraints, e.g., chance constraints and conditional Value-at-Risk constraints. This paper studies these two types of risk constraints when the probability distribution of the uncertain parameters is ambiguous. In particular, we assume that the distributional information consists of the first two moments of the uncertainty and a generalized notion of unimodality. We find that the ambiguous risk constraints in this setting can be recast as a set of second-order cone (SOC) constraints. In order to facilitate the algorithmic implementation, we also derive efficient ways of finding violated SOC constraints. Finally, we demonstrate the theoretical results via computational case studies on power system operations.


Ambiguity Chance constraints Conditional Value-at-Risk Second-order cone representation Separation Golden section search 

Mathematics Subject Classification

90C15 Stochastic programming 90C22 Semidefinite programming 90C34 Semi-infinite programming 



This research has been supported in part by the National Science Foundation (NSF) under Grants CMMI-1555983 and CCF-1442495.


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Copyright information

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

Authors and Affiliations

  1. 1.Department of Electrical Engineering and Computer ScienceUniversity of MichiganAnn ArborUSA
  2. 2.Department of Industrial and Operations EngineeringUniversity of MichiganAnn ArborUSA

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