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 -