Stochastic linear programming with a distortion risk constraint

TL;DR

This paper develops a method for solving stochastic linear programming problems with a distortion risk constraint, using a geometrical algorithm and analyzing its asymptotic behavior, with implementation in an R-package.

Contribution

It introduces a novel geometrical algorithm for stochastic linear programs with a distortion risk constraint and studies its asymptotic properties.

Findings

Algorithm efficiently solves the problem

Uncertainty sets are convex polytopes

Asymptotic analysis confirms robustness

Abstract

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.