Journal
MATHEMATICAL METHODS OF OPERATIONS RESEARCH
Volume 71, Issue 3, Pages 535-549Publisher
SPRINGER HEIDELBERG
DOI: 10.1007/s00186-010-0316-3
Keywords
Stochastic programming; Probabilistic constraints; Chance constraints; Derivative of probabilities of rectangles; Water reservoir management
Funding
- OSIRIS Department of Electricite de France RD
- DFG Research Center Matheon Mathematics for key technologies in Berlin
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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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