Stochastic linear programming with a distortion risk constraint

Linear optimization problems are investigated whose parameters are uncertain. We apply coherent distortion risk measures to capture the possible violation of a restriction. Each risk constraint induces an uncertainty set of coefficients, which is shown to be a weighted-mean trimmed region. Given an external sample of the coefficients, an uncertainty set is a convex polytope that can be exactly calculated. We construct an efficient geometrical algorithm to solve stochastic linear programs that have a single distortion risk constraint. The algorithm is available as an R-package. Also the algorithm’s asymptotic behavior is investigated, when the sample is i.i.d. from a general probability distribution. Finally, we present some computational experience. Subject classifications: Programming: stochastic. Decision analysis: risk. Statistics: nonparametric.