Quantitative stability in stochastic programming: The method of probability metrics

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976)

MSC 2000

90C15 Stochastic programming

90C31 Sensitivity, stability, parametric optimization

60E05 Distributions: general theory

60B10 Convergence of probability measures

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.

Keywords: Stochastic programming, quantitative stability, probability metrics, Fortet-Mourier metrics, empirical approximations, two-stage models, chance constrained models, stable portfolio models