Risk-Averse Stochastic Modeling and Optimization

Abstract

The ability to compare random outcomes based on the decision makers' risk preferences is crucial to modeling decision-making problems under uncertainty. In this tutorial, the primary focus is on the stochastic preference relations based on the widely applied risk measure, conditional value-at-risk (CVaR), and the second-order stochastic dominance (SSD). We present single- and two-stage stochastic optimization problems that feature such risk-averse preference relations. We discuss the main computational challenges in solving the problems of interest, and for finite probability spaces, we describe alternative mathematical programming formulations and effective solution methods. Our focus is on delayed cut generation solution algorithms, which rely on a Benders-type scenario decomposition approach in the case of a two-stage problem. In addition, we review the recent developments in risk-averse stochastic programming, with a particular focus on multicriteria optimization problems that feature multivariate stochastic benchmarking constraints based on CVaR and SSD.