2009-10-16Buch DOI: 10.18452/3029
On probabilistic constraints induced by rectangular sets and multivariate normal distributions
Ackooij, W. Van
In this paper, we consider optimization problems under probabilistic constraints which are deﬁned by two-sided inequalities for the underlying normally distributed random vector. As a main step for an algorithmic solution of such problems, we derive a derivative formula for (normal) probabilities of rectangles as functions of their lower or upper bounds. This formula allows to reduce the calculus of such derivatives to the calculus of (normal) probabilities of rectangles themselves thus generalizing a similar well-known statement for multivariate normal distribution functions. As an application, we consider a problem from water reservoir management. One of the outcomes of the problem solution is that the (still frequently encountered) use of simple individual probabilistic can completely fail. In contrast, the (more diﬃcult) use of joint probabilistic constraints which heavily depends on the derivative formula mentioned before yields very reasonable and robust solutions over the whole time horizon considered.
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