TY - JOUR

T1 - Quantitative stability in stochastic programming

T2 - The method of probability metrics

AU - Rachev, Svetlozar T.

AU - Römisch, Werner

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2002/11

Y1 - 2002/11

N2 - 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.

AB - 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.

KW - Chance-constrained models

KW - Empirical approximations

KW - Fortet-Mourier metrics

KW - Mixed-integer

KW - Probability metrics

KW - Quantitative stability

KW - Stable portfolio models

KW - Stochastic programming

KW - Two-stage models

UR - http://www.scopus.com/inward/record.url?scp=0036873891&partnerID=8YFLogxK

U2 - 10.1287/moor.27.4.792.304

DO - 10.1287/moor.27.4.792.304

M3 - Article

AN - SCOPUS:0036873891

VL - 27

SP - 792

EP - 818

JO - Mathematics of Operations Research

JF - Mathematics of Operations Research

SN - 0364-765X

IS - 4

ER -