Linear Programming with Probability Constraints - Part 1

In this paper, we discuss linear programs in which the data that specify the constraints are subject to random uncertainty. A usual approach in this setting is to enforce the constraints up to a given level of probability. We show that for a wide class of probability distributions (i.e. radial distributions) on the data, the probability constraints can be explicitly converted into convex second order cone (SOC) constraints, hence the probability constrained linear program can be solved exactly with great efficiency. We next analyze the situation when the probability distribution of the data in not completely specified, but it is only known to belong to a given class of distributions. In this case, we provide explicit convex conditions that guarantee the satisfaction of the probability constraints, for any possible distribution belonging to the given class.