Coherent risk measures have become a popular tool for incorporating risk aversion into stochastic optimization models. For dynamic models in which un-certainly is resolved at more than one stage, however, use of coherent risk measures within a standard single-level optimization framework presents the modeler with an uncomfortable choice between two desirable model properties, time consistency and law invariance. Prior published work has favored maintaining time consistency, but the absence of law invariance makes the resulting models unattractive to practical decision makers. This paper summarizes these issues and then presents an alternative multilevel optimization modeling approach that preserves law invariance, yet leads to models that are time-consistent even while using time-inconsistent risk measures. It argues that this approach should be the starting point for all multistage optimization modeling; however, when performing classical risk-neutral modeling, it simplifies to a more familiar single-objective form. Unfortunately, this paper also shows that its proposed approach leads to NP-hard models, even in the simplest imaginable setting in which it would be needed: three-stage linear problems on a nite probability space, using the standard mean-semideviation and average-value-at-risk measures. While not necessarily indicating that solution of such models is impractical, these results suggest that it will likely require approximation or implicit enumeration methods. We close with some preliminary computational results showing that high-quality local optimum solutions of models of the kind we propose are in fact practically computable, hence that the complexity results should not be taken as completely discouraging.
RUTCOR Research Report RRR 7-2013, Rutgers University August 2013
View Time Consistency Versus Law Invariance in Multistage Stochastic Optimization with Coherent Risk Measures: Multilevel Optimization Modeling and Computational Complexity