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

TL;DR

This paper introduces an analytical, problem-driven scenario generation method tailored for stochastic programs with tail risk measures, focusing on accurately capturing tail behavior to improve decision-making in portfolio and network design problems.

Contribution

It proposes a novel analytical approach that targets tail regions of distributions for scenario generation, enhancing representation of tail risk in stochastic programming.

Findings

Better representation of tail risk in scenarios.

More stable solutions in portfolio selection.

Improved performance over standard Monte Carlo methods.

Abstract

Scenario generation is the construction of a discrete random vector to represent parameters of uncertain values in a stochastic program. Most approaches to scenario generation are distribution-driven, that is, they attempt to construct a random vector which captures well in a probabilistic sense the uncertainty. On the other hand, a problem-driven approach may be able to exploit the structure of a problem to provide a more concise representation of the uncertainty. In this paper we propose an analytic approach to problem-driven scenario generation. This approach applies to stochastic programs where a tail risk measure, 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 scenarios evenly across the support of the random vector, struggle to adequately represent tail risk. Our scenario generation approach works by targeting the construction of scenarios in areas of the distribution corresponding to the tails of the loss distributions. We provide conditions under which our approach is consistent with sampling, and as proof-of-concept demonstrate how our approach could be applied to two classes of problem, namely network design and portfolio selection. Numerical tests on the portfolio selection problem demonstrate that our approach yields better and more stable solutions compared to standard Monte Carlo sampling.