Approximations for and Convexity of Probabilistically Constrained Problems with Random Right-hand Sides

We consider probabilistically constrained problems, in which the multivariate random variables are located in the right-hand sides. The objective function is linear, and its optimization is subject to a set of linear constraints as well as a joint probabilistic constraint enforcing that the joint fulfillment of a system of linear inequalities with random right-hand side variables be above a prescribed probability level p. To deal with such complex problems, we describe a solving method based on the pefficiency concept for discretely distributed random variables, and also propose some alternative formulations applicable to both discrete and continuous probability distributions, and involving the substitution of the joint probabilistic constraint by a set of individual constraints, the Boole’s inequality, the binomial moment bounding scheme, and Slepian’s inequality, respectively. The common advantage of these formulations is that they involve the computation of joint probabilistic constraints of lower dimension than this of the joint probabilistic constraint included in the original formulation. We analyze their computational tractability, and evaluate their constraining power relying on three datasets, in which random variables have a normal distribution. We then prove that the function [ 0] i i E T x T x ε ε − − > enforcing a reliability level di is concave except for very large (small) values of Tix (di). We study the relationship between the service levels pi and di defined by the functions and ( ) i i i P T x p ε ≥ ≥ and [ 0] i i i i i E T x T x d ε ε − − > ≤ , and use the STABIL problem to provide a power management interpretation of the results.