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Scenario generation is the construction of a discrete random vector torepresent parameters of uncertain values in a stochastic program. Mostapproaches to scenario generation are distribution-driven, that is, theyattempt to construct a random vector which captures well in a probabilisticsense the uncertainty. On the other hand, a problem-driven approach may be ableto exploit the structure of a problem to provide a more concise representationof the uncertainty.  In this paper we propose an analytic approach to problem-driven scenariogeneration. This approach applies to stochastic programs where a tail riskmeasure, such as conditional value-at-risk, is applied to a loss function.Since tail risk measures only depend on the upper tail of a distribution,standard methods of scenario generation, which typically spread their scenariosevenly across the support of the random vector, struggle to adequatelyrepresent tail risk. Our scenario generation approach works by targeting theconstruction of scenarios in areas of the distribution corresponding to thetails of the loss distributions. We provide conditions under which our approachis consistent with sampling, and as proof-of-concept demonstrate how ourapproach could be applied to two classes of problem, namely network design andportfolio selection. Numerical tests on the portfolio selection problemdemonstrate that our approach yields better and more stable solutions comparedto standard Monte Carlo sampling.

Problem-driven scenario generation: an analytical approach for stochastic programs with tail risk measure