A Second-Order Sufficient Optimality Condition for Risk-Neutral Bi-level Stochastic Linear Programs

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.

Automated summary

The text discusses the gradient of the expectation functional in risk-neutral bi-level stochastic linear models with random lower-level right-hand side. It shows that the gradient may not be locally Lipschitz continuous under certain assumptions, but provides sufficient conditions for Lipschitz continuity. The text also studies geometric properties of regions of strong stability and derivation results that may aid in gradient computation.