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.
Highlights
Stochastic programming is a tool for making decisions under uncertainty
We propose an analytic problem-driven approach to scenario generation applicable to stochastic programs which use tail risk measures of a form made precise in Sect
This paper is organized as follows: in Sect. 2 we discuss related work; in Sect. 3 we define tail risk measures and their associated risk regions; in Sect. 4 we discuss how these risk regions can be exploited for the purposes of scenario generation; in Sect. 5 we prove that our scenario generation method is consistent with standard Monte Carlo sampling; in Sects. 6 and 7 we derive risk regions for the two classes of problems described above; in Sect. 8 we present numerical tests; in Sect. 9 we summarize our results and make some concluding remarks
Summary
Stochastic programming is a tool for making decisions under uncertainty. Under this modeling paradigm, uncertain parameters are modeled as a random vector, and one Typically, a stochastic program can only be solved when it is scenario-based, that is when the random vector for the problem has a finite discrete distribution. Stochastic linear programs become large-scale linear programs when the underlying random vector is discrete. In the stochastic programming literature, the mass points of this random vector are referred to as scenarios, the discrete distribution as the scenario set and the construction of this set as scenario generation. Scenario generation can consist of discretizing a continuous probability distribution, or directly modeling the uncertain quantities as discrete random variables. The key issue of scenario generation is how to represent the uncertainty to ensure that the solution to the problem is reliable, while keeping the number of scenarios low so that the problem is computationally tractable
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