Ambiguous chance constrained problems and robust optimization

Abstract

In this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain. We focus primarily on the special case where the uncertainty set of the distributions is of the form where ρ p denotes the Prohorov metric. The ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure The main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with a high probability. This result is established using the Strassen-Dudley Representation Theorem that states that when the distributions of two random variables are close in the Prohorov metric one can construct a coupling of the random variables such that the samples are close with a high probability. We also show that the robust sampled problem can be solved efficiently both in theory and in practice.

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Correspondence to G. Iyengar.

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Research partially supported by NSF grant CCR-00-09972.

Research partially supported by NSF grants CCR-00-09972, DMS-01-04282, and ONR grant N000140310514.

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Erdoğan, E., Iyengar, G. Ambiguous chance constrained problems and robust optimization. Math. Program. 107, 37–61 (2006). https://doi.org/10.1007/s10107-005-0678-0

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Keywords

  • Robust optimization
  • Stochastic programming
  • Learning Theory
  • Coupling of random variables
  • Ambiguity in measure
  • Sample approximation
  • VC dimension

Mathematics Subject Classification (2000)

  • 62G35
  • 90C15
  • 68Q32