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Scenario reduction for stochastic programs with Conditional Value-at-Risk

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Abstract

In this paper we discuss scenario reduction methods for risk-averse stochastic optimization problems. Scenario reduction techniques have received some attention in the literature and are used by practitioners, as such methods allow for an approximation of the random variables in the problem with a moderate number of scenarios, which in turn make the optimization problem easier to solve. The majority of works for scenario reduction are designed for classical risk-neutral stochastic optimization problems; however, it is intuitive that in the risk-averse case one is more concerned with scenarios that correspond to high cost. By building upon the notion of effective scenarios recently introduced in the literature, we formalize that intuitive idea and propose a scenario reduction technique for stochastic optimization problems where the objective function is a Conditional Value-at-Risk. Numerical results presented with problems from the literature illustrate the performance of the method and indicate the cases where we expect it to perform well.

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Notes

  1. Models with CVaR have also been used in the context of multistage stochastic programs, but we do not review them here as we focus on the two-stage case in this paper.

  2. By “reducing” the number of scenarios we mean finding another distribution \(\hat{P}\) such that the support of \(\hat{P}\) is strictly contained in \(\Xi \) (the support of \(\hat{P}\) consists of those scenarios \(\xi _i\) for which \(\hat{P}_i>0\)).

  3. In the most general case W and q are random as well, but for our purposes we assume that these quantities are deterministic.

  4. An implementation of ZEBRA can be found in https://sites.google.com/site/sergiogarciaquiles/the-p-median-facility-location-problem.

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Acknowledgements

We thank the anonymous referees for their constructive comments which helped improve the presentation of our results. This work has been supported by FONDECYT 1171145, Chile.

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  1. School of Business, Universidad Adolfo Ibañez, Santiago, Chile

    Sebastián Arpón, Tito Homem-de-Mello & Bernardo Pagnoncelli

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  1. Sebastián Arpón
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  2. Tito Homem-de-Mello
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Correspondence to Tito Homem-de-Mello.

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Arpón, S., Homem-de-Mello, T. & Pagnoncelli, B. Scenario reduction for stochastic programs with Conditional Value-at-Risk. Math. Program. 170, 327–356 (2018). https://doi.org/10.1007/s10107-018-1298-9

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