Mathematical Programming

, Volume 107, Issue 1, pp 37–61

Ambiguous chance constrained problems and robust optimization

Article

DOI: 10.1007/s10107-005-0678-0

Cite this article as:
Erdoğan, E. & Iyengar, G. Math. Program. (2006) 107: 37. doi:10.1007/s10107-005-0678-0

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 Open image in new window of the distributions is of the form Open image in new window where ρp 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 Open image in new window 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.

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 

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.IEOR DepartmentColumbia UniversityNew YorkUSA

Personalised recommendations