Concavity and efficient points of discrete distributions in probabilistic programming

We consider stochastic programming problems with probabilistic constraints involving integer-valued random variables. The concept of a p-efficient poi nt of a probability distribution is used to derive various equivalent probl em formulations. Next we introduce the concept of r-concave discrete probab ility distributions and analyse its relevance for problems under considerat ion. These notions are used to derive lower and upper bounds for the optima l Value of probabilistically constrained stochastic programming problems wi th discrete random variables. The results are illustrated with numerical ex amples.