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Quantitative stability in stochastic programming: The method of probability metrics
, 2000
"... 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 minim ..."
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Cited by 27 (12 self)
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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 specic models, namely for linear twostage, mixedinteger twostage 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 underly...
Metric Regularity and Quantitative Stability in Stochastic Programs With Probabilistic Constraints
"... Introducing probabilistic constraints leads in general to nonconvex, nonsmooth or even discontinuous optimization models. In this paper, necessary and sufficient conditions for metric regularity of (several joint) probabilistic constraints are derived using recent results from nonsmooth analysis. Th ..."
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Cited by 12 (9 self)
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Introducing probabilistic constraints leads in general to nonconvex, nonsmooth or even discontinuous optimization models. In this paper, necessary and sufficient conditions for metric regularity of (several joint) probabilistic constraints are derived using recent results from nonsmooth analysis. The conditions apply to fairly general constraints and extend earlier work in this direction. Further, a verifiable sufficient condition for quadratic growth of the objective function in a more specific convex stochastic program is indicated and applied in order to obtain a new result on quantitative stability of solution sets when the underlying probability distribution is subjected to perturbations. This is used to derive bounds for the deviation of solution sets when the probability measure is replaced by empirical estimates. Keywords: stochastic programming, probabilistic constraints, metric regularity, nonsmooth analysis, quadratic growth, quantitative stability, empirical approximation ...
Stability of Solutions to Chance Constrained Stochastic Programs
"... Perturbations of convex chance constrained stochastic programs are considered the underlying probability distributions of which are rconcave. Verifiable sufficient conditions are established guaranteeing Holder continuity properties of solution sets with respect to variations of the original distri ..."
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Cited by 3 (1 self)
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Perturbations of convex chance constrained stochastic programs are considered the underlying probability distributions of which are rconcave. Verifiable sufficient conditions are established guaranteeing Holder continuity properties of solution sets with respect to variations of the original distribution. Examples illustrate the potential, sharpness and limitations of the results.
Math. Program., Ser. A 100: 589–611 (2004) Digital Object Identifier (DOI) 10.1007/s101070040507x
, 2004
"... Abstract. We study perturbations of a stochastic program with a probabilistic constraint and rconcave original probability distribution. First we improve our earlier results substantially and provide conditions implying Hölder continuity properties of the solution sets w.r.t. the Kolmogorov distanc ..."
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Abstract. We study perturbations of a stochastic program with a probabilistic constraint and rconcave original probability distribution. First we improve our earlier results substantially and provide conditions implying Hölder continuity properties of the solution sets w.r.t. the Kolmogorov distance of probability distributions. Secondly, we derive an upper Lipschitz continuity property for solution sets under more restrictive conditions on the original program and on the perturbed probability measures. The latter analysis applies to linearquadratic models and is based on work by Bonnans and Shapiro. The stability results are illustrated by numerical tests showing the different asymptotic behaviour of parametric and nonparametric estimates in a program with a normal probabilistic constraint. Key words. probabilistic constraints – chance constraints – Lipschitz stability – stochastic optimization 1.