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Jonathan Eckstein(jecksteirci.rutgers.edu) Abstract: Coherent risk measures have become a popular tool for incorporating risk aversion into stochastic optimization models. For dynamic models in which uncertainty is resolved at more than one stage, however, using coherent risk measures within a standard singlelevel optimization framework becomes problematic. To avoid severe timeconsistency difficulties, the current state of the art is to employ risk measures of a specific nested form, which unfortunately have some undesirable and somewhat counterintuitive modeling properties. For one thing, this technique requires increasing risk aversion as risks and reward recede in time. Further, it produces objective functions that cannot be law invariant with respect to the total incurred costs and rewards, meaning that two solutions with identical probability distributions of final wealth may be assigned different levels of risk, and the nested form of the objective function cannot be simplified. These properties deter practical acceptance of such models, and are particularly undesirable for situations with close final time horizons. This paper summarizes these issues and then presents an alternative multilevel optimization modeling approach that enforces a form of time consistency through constraints rather than by restricting the modeler's choice of objective function. This technique leads to models that are timeconsistent even while using timeinconsistent risk measures, and can easily be formulated to be law invariant with respect to the final wealth if so desired. We argue that this approach should be the starting point for all multistage optimization modeling. When used with timeconsistent objective functions, we show its multilevel optimization constraints become redundant and the associated models thus simplify to a more familiar singleobjective form. Unfortunately, this paper also shows that its proposed approach leads to NPhard models, even in the simplest imaginable setting in which it would be needed: threestage linear problems on a finite probability space, using the standard meansemideviation and averagevalueatrisk measures. Finally, we show that for a simple but reasonably realistic test application, that the kind of models we propose, while being drawn from an NPhard family and certainly more time consuming to solve than those obtained from the nestedobjective approach, are readily solvable to global optimality using a standard commercial MILP solver. Therefore, there seems some promise of the modeling approach being useful despite its computational complexity properties. Keywords: Stochastic programming, risk measures, computational complexity, multilevel optimization Category 1: Stochastic Programming Citation: Download: [PDF] Entry Submitted: 08/24/2014 Modify/Update this entry  
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