Abstract
The expectation functionals, which arise in risk-neutral bi-level stochastic linear models with random lower-level right-hand side, are known to be continuously differentiable, if the underlying probability measure has a Lebesgue density. We show that the gradient may fail to be local Lipschitz continuous under this assumption. Our main result provides sufficient conditions for Lipschitz continuity of the gradient of the expectation functional and paves the way for a second-order optimality condition in terms of generalized Hessians. Moreover, we study geometric properties of regions of strong stability and derive representation results, which may facilitate the computation of gradients.
Highlights
We study bi-level stochastic linear programs with random right-hand side in the lowerlevel constraint system
We show that the assumptions of [1] are too weak to even guarantee local Lipschitz continuity of the gradient
As part of the preparatory work for the proof of the main result, we in particular show that any region of strong stability in the sense of [1, Definition 4.1] is a finite union of polyhedral cones
Summary
We study bi-level stochastic linear programs with random right-hand side in the lowerlevel constraint system. The main result of the present work provides sufficient conditions, namely boundedness of the support and uniform boundedness of the Lebesgue density of the underlying probability measure, that ensure Lipschitz continuity of the gradient of the expectation functional. As part of the preparatory work for the proof of the main result, we in particular show that any region of strong stability in the sense of [1, Definition 4.1] is a finite union of polyhedral cones This representation is of independent interest, as it may facilitate the calculation or estimation of gradients of the expectation functional and enhance gradient descent-based approaches. Results of these sections play an important role in the proof of the main result that is given in Sect.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
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.
