Quantitative stability in stochastic programming: The method of probability metrics

Svetlozar T. Rachev, Werner Römisch

Research output: Contribution to journalArticlepeer-review

94 Scopus citations


Quantitative stability of optimal values and solution sets to stochastic programming problems is studied when the underlying probability distribution varies in some metric space of probability measures. We give conditions that imply that a stochastic program behaves stable with respect to a minimal information (m.i.) probability metric that is naturally associated with the data of the program. Canonical metrics bounding the m.i. metric are derived for specific models, namely for linear two-stage, mixed-integer two-stage and chance-constrained models. The corresponding quantitative stability results as well as some consequences for asymptotic properties of empirical approximations extend earlier results in this direction. In particular, rates of convergence in probability are derived under metric entropy conditions. Finally, we study stability properties of stable investment portfolios having minimal risk with respect to the spectral measure and stability index of the underlying stable probability distribution.

Original languageEnglish
Pages (from-to)792-818
Number of pages27
JournalMathematics of Operations Research
Issue number4
StatePublished - Nov 2002


  • Chance-constrained models
  • Empirical approximations
  • Fortet-Mourier metrics
  • Mixed-integer
  • Probability metrics
  • Quantitative stability
  • Stable portfolio models
  • Stochastic programming
  • Two-stage models


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