On probabilistic constraints induced by rectangular sets and multivariate normal distributions

Abstract

In this paper, we consider optimization problems under probabilistic constraints which aredefined by two-sided inequalities for the underlying normally distributed random vector. Asa 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 formulaallows to reduce the calculus of such derivatives to the calculus of (normal) probabilitiesof rectangles themselves thus generalizing a similar well-known statement for multivariatenormal distribution functions. As an application, we consider a problem from water reservoirmanagement. One of the outcomes of the problem solution is that the (still frequentlyencountered) 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.