On probabilistic constraints induced by rectangular sets and multivariate normal distributions
(2009)
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.
In this paper, we deal with a hydraulic reservoir optimization problem with uncertainty on
inflows in a joint chance constrained programming setting. In particular, we will consider inflows with
a persistency effect, following a causal time series model, and examine the impact of the ”Gaussian”
assumption for such inflows. We present an iterative algorithm for solving similarly structured joint
chance constrained programming problems that requires a Slater point and the computation of gradients.
Several alternatives to the joint chance constraint problem are presented. In particular, we present an
individual chance constraint problem and a robust model. We illustrate the interest of joint chance
constrained programming by comparing results obtained on a realistic hydro-valley with those obtained
from the alternative models. Despite the fact that the alternative models often require less hypothesis
on the law of the inflows, we show that they yield conservative and costly solutions. The simpler models,
such as the individual chance constraint one, are shown to yield insufficient robustness and are therefore
not useful. We therefore conclude that Joint Chance Constrained programming appears as a technique
offering a good trade-off between cost and robustness and can be tractable for complex realistic models.