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.

中文翻译：

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风险中性双层随机线性规划的二阶充分最优条件

如果潜在概率测度具有 Lebesgue 密度，则在具有随机低级右侧的风险中性双级随机线性模型中出现的期望函数已知是连续可微的。我们表明在这个假设下梯度可能不能是局部 Lipschitz 连续的。我们的主要结果为期望泛函梯度的 Lipschitz 连续性提供了充分条件，并为广义 Hessians 的二阶最优性条件铺平了道路。此外，我们研究了强稳定性区域的几何特性并得出表示结果，这可能有助于梯度的计算。