Stochastic linear programming with a distortion risk constraint

Coherent distortion risk measures are applied to capture the possible violation of a restriction in linear optimization problems whose parameters are uncertain. Each risk constraint induces an uncertainty set of coefficients, which is proved to be a weighted-mean trimmed region. Thus, given a 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. The algorithm’s asymptotic behavior is also investigated, when the sample is i.i.d. from a general probability distribution. Finally, we present some computational experience.