Probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra

We consider stochastic programming problems with probabilistic constraints in volving random variables with discrete distributions They can be reformulated as large scale mixed integer programming problems with knapsack constraints Using speci c properties of stochastic programming problems and bounds on the prob ability of the union of events we develop new inequalities for these mixed integer programs We also develop methods for lifting these inequalities These procedures are used in a general iterative algorithm for solving probabilistically constrained problems The results are illustrated with a numerical example