Distributionally robust optimization with polynomial densities: theory, models and algorithms

  • Etienne de Klerk
  • Daniel KuhnEmail author
  • Krzysztof Postek
Full Length Paper Series B


In distributionally robust optimization the probability distribution of the uncertain problem parameters is itself uncertain, and a fictitious adversary, e.g., nature, chooses the worst distribution from within a known ambiguity set. A common shortcoming of most existing distributionally robust optimization models is that their ambiguity sets contain pathological discrete distributions that give nature too much freedom to inflict damage. We thus introduce a new class of ambiguity sets that contain only distributions with sum-of-squares (SOS) polynomial density functions of known degrees. We show that these ambiguity sets are highly expressive as they conveniently accommodate distributional information about higher-order moments, conditional probabilities, conditional moments or marginal distributions. Exploiting the theoretical properties of a measure-based hierarchy for polynomial optimization due to Lasserre (SIAM J Optim 21(3):864–885, 2011), we prove that certain worst-case expectation constraints are polynomial-time solvable under these new ambiguity sets. We also show how SOS densities can be used to approximately solve the general problem of moments. We showcase the applicability of the proposed approach in the context of a stylized portfolio optimization problem and a risk aggregation problem of an insurance company.


Distributionally robust optimization Semidefinite programming Sum-of-squares polynomials Generalized eigenvalue problem 

Mathematics Subject Classification

90C22 90C26 90C15 



Etienne de Klerk would like to thank Dorota Kurowicka and Jean Bernard Lasserre for valuable discussions and references. Daniel Kuhn gratefully acknowledges financial support from the Swiss National Science Foundation under grant BSCGI0_157733.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.Tilburg UniversityTilburgThe Netherlands
  2. 2.EPFLLausanneSwitzerland
  3. 3.Erasmus University RotterdamRotterdamThe Netherlands

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