Decomposition of chance constraint optimization problem

Question

I want to decompose a chance constraint optimization problem and the constraint is:

$Pr\left( \sum_{i}^{}{\left( x_{i}+\xi _{i} \right)}\leq c \right)\geq 1-\epsilon $

where $\xi _{i}$ are independent random variables with known distributions.

I think the calculation of the $\sum_{i}^{}{\xi _{i}}\leq z$ is not computable because $i$ is so large.

So I wondering if it is possible to decompose the constraint into separated ones even with some reasonable approximation, so that I can decompose the optimization problem?

e.g.

$Pr\left( x_{1}\leq b_{1} \right)\geq 1-\epsilon _{1}\; $
$Pr\left( x_{2}\leq b_{2} \right)\geq 1-\epsilon _{2}\; $ ...

Answer 1

I think I can use the approximation method in the following paper to turn the chance constraint to a deterministic constraint and then use dual decomposition.

http://www.researchgate.net/publication/220589607_Selected_topics_in_robust_convex_optimization/file/72e7e51b5af0cedfa8.pdf