On probabilistic constraints with multivariate truncated Gaussian and lognormal distributions
(2016)
Many engineering problems with uncertain data, notably arising
in power management, can be formulated as optimization problems subject to
probabilistic constraints. While dealing with such constraints under continuous distributions of the underlying random parameter remains a difficult task
in general both from the numerical and theoretical point of view, quite some
progress has been made in the special case of multivariate Gaussian distributions. These are not perfectly adequate, however, in many circumstances, in
particular not, when modeling uncertain inflows to hydro reservoirs or uncertain demands in gas networks. Interesting alternatives are offered by truncations of multivariate Gaussian distributions to polyhedra or by multivariate
lognormal distributions. The paper discusses the applicability of such distributions in the context of a simple joint linear probabilistic constraint putting
the emphasis on the numerical approximation of probabilities and their gradients (w.r.t. decisions to be optimized) as well as on the convexity of the set
of feasible decisions.