On probabilistic constraints with multivariate truncated Gaussian and lognormal distributions

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.