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Quantitative Stability in Stochastic Programming: The Method of Probability Metrics

Svetlozar T. Rachev and Werner Römisch
Mathematics of Operations Research
Vol. 27, No. 4 (Nov., 2002), pp. 792-818
Published by: INFORMS
Stable URL: http://www.jstor.org/stable/3690468
Page Count: 27
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Quantitative Stability in Stochastic Programming: The Method of Probability Metrics
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Abstract

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.

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