On probabilistic constraints induced by rectangular sets and multivariate normal distributions |
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Authors: | Wim Van Ackooij René Henrion Andris Möller Riadh Zorgati |
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Institution: | 1. électricité de France R&D, 1, avenue du Général de Gaulle, 92141, Clamart CEDEX, France 2. Weierstrass Institute, Mohrenstra?e 39, 10117, Berlin, Germany
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Abstract: | 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 prove
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 constraints can completely fail. By 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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