Approximation of Probabilistic Constraints in Stochastic Programming Problems with a Probability Measure Kernel

Abstract

We consider a linear stochastic programming problem with a deterministic objective function and individual probabilistic constraints. Each probabilistic constraint is a lower bound on the probability function equal to the probability of the fulfillment of a certain linear inequality. We propose to first represent probabilistic constraints in the form of equivalent inequalities for the quantile functions. After that, each quantile function is approximated using the confidence method. The main analytic tool is based on polyhedral approximation of the p-kernel for the multidimensional probability distribution. For the case when probability functions are defined by linear inequalities, constraints on quantile functions are with arbitrary accuracy approximated by systems of deterministic linear inequalities. As a result, the original problem is approximated by a linear programming problem.