Probability functions appearing in chance constraints are an ingredient of many practical applications. Understanding differentiability, and providing explicit formulae for gradients, allow us to build nonlinear programming methods for solving these optimization problems from practice. Unfortunately, differentiability of probability functions cannot be taken for granted. In this paper, motivated by gas network applications, we investigate differentiability of probability functions acting on non-convex quadratic forms. We establish continuous differentiability for the broad class of elliptical random vectors under mild conditions.

中文翻译：

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非凸二次型非线性概率约束的梯度公式

出现在机会约束中的概率函数是许多实际应用的组成部分。了解可微性并提供显式的梯度公式，使我们能够构建非线性规划方法来从实践中解决这些优化问题。不幸的是，不能想当然地认为概率函数的可微性。在本文中，受气体网络应用的启发，我们研究了作用于非凸二次形式的概率函数的可微性。我们在温和条件下为广泛的椭圆随机向量建立了连续可微性。