Abstract
This paper investigates the stability of optimal solution sets to stochastic programs with complete recourse, where the underlying probability measure is understood as a parameter varying in some space of probability measures. In (Math. Programming, 67 (1994), pp. 99-108) Sha- piro proved Lipschitz upper semicontinuity of the solution set mapping. Inspired by this result, we introduce a subgradient distance for probability distributions and establish the persistence of optimal solutions. For a subclass of recourse models we show that the solution set mapping is (Hausdorff) Lipschitz continuous with respect to the subgradient distance. Moreover, the subgradient distance is estimated above by the Kolmogorov-Smirnov distance of certain distribution functions related to the recourse model. The Lipschitz continuity result is illustrated by verifiable sufficient conditions for stochastic programs to belong to the mentioned subclass and by examples showing its validity and limitations. Finally, the Lipschitz continuity result is used to derive some new results on the asymptotic behavior ofoptimal solutions when the probability measure underlying the recourse model is estimated via empirical measures (law of iterated logarithm, large deviation estimate, estimate for asymptotic distribution).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
