On probabilistic constraints induced by rectangular sets and multivariate normal distributions

W. Van Ackooij; R. Henrion; A. Möller; R. Zorgati

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SPEPS Preprint

Author(s):	W. Van Ackooij, Électicité de France R&D R. Henrion, WIAS A. Möller, WIAS R. Zorgati, Électricité de France R&D
Title:	On probabilistic constraints induced by rectangular sets and multivariate normal distributions
Date of Acceptance:	16.10.2009
Submission Date:	01.09.2009
Series Title:	Stochastic Programming E-Print Series (SPEPS)
Editors:	Julie L. Higle; Werner Römisch; Surrajeet Sen
Complete Preprint:	pdf (urn:nbn:de:kobv:11-100101196)
Keywords (eng):	stochastic programming, probabilistic constraints, derivative of probabilities of rectangles, water reservoir management
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Abstract (eng):

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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