In stochastic programming (SP), a common constraint is of the form $$ \mathbb{P}(f(x,\xi) \leq 0) \geq \alpha. $$ If \(f\) is affine in \(x\) and \(\xi\), then we have the chance constraint of the form $$ \mathbb{P}(A(\xi)x + b(\xi)+ c \leq 0) \geq \alpha $$ for \(A\in \mathbb{R}^{m \times n} \). In this case, we can represent the joint constraint with \(m\) individual constraints of the form $$ \mathbb{P}(a_i^T(\xi)x + b_i(\xi) + c \leq 0) \geq \alpha. $$ To confirm, we can do this because each dimension is independent? How can I better see this? Also, if \(f\) were not linear in \(x\) and \(\xi\), then we are unable to separate the joint constraint into constraint components, right? A simple, introductory reference on this idea would be much appreciated!
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