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Risk-averse feasible policies for large-scale multistage stochastic linear programs

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Abstract

We consider risk-averse formulations of stochastic linear programs having a structure that is common in real-life applications. Specifically, the optimization problem corresponds to controlling over a certain horizon a system whose dynamics is given by a transition equation depending affinely on an interstage dependent stochastic process. We put in place a rolling-horizon time consistent policy. For each time step, a risk-averse problem with constraints that are deterministic for the current time step and uncertain for future times is solved. To each uncertain constraint corresponds both a chance and a Conditional Value-at-Risk constraint. We show that the resulting risk-averse problems are numerically tractable, being at worst conic quadratic programs. For the particular case in which uncertainty appears only on the right-hand side of the constraints, such risk-averse problems are linear programs. We show how to write dynamic programming equations for these problems and define robust recourse functions that can be approximated recursively by cutting planes. The methodology is assessed and favourably compared with Stochastic Dual Dynamic Programming on a real size water-resource planning problem.

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Notes

  1. The authors specially acknowledge the good will and availability of Débora Dias Jardim Penna.

  2. We adopt the convention 1 MWMonth \(=\frac{365.25{\small {\times }}24}{12}\) MWh \(=730.5\) MWh.

  3. If \(X\) is a continuous random variable for which lower values are preferred, its Value-at-Risk of level \({\varepsilon _{\scriptstyle \mathtt{p }}}\) is given by \(\textit{VaR}_{{\varepsilon _{\scriptstyle \mathtt{p }}}}(X):=F_{X}^{-1}(1-{\varepsilon _{\scriptstyle \mathtt{p }}})\) for any \({\varepsilon _{\scriptstyle \mathtt{p }}}\in [0,1]\).

    Table 1 Central and dispersion characteristics of the total cost (R$)
    Full size table

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Acknowledgments

We would like to thank the three reviewers and the Associate Editor for beneficial comments and suggestions.

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Authors and Affiliations

  1. Escola de Matemática Aplicada, Fundação Getulio Vargas, praia de Botafogo, Rio de Janeiro, Brazil

    Vincent Guigues

  2. Departamento de Engenharia Industrial, Escola Politécnica, UFRJ, Ilha do Fundão, CT, Bloco F, Rio de Janeiro, Brazil

    Vincent Guigues

  3. IMPA – Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ 22460-320, Brazil

    Vincent Guigues & Claudia Sagastizábal

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  1. Vincent Guigues
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  2. Claudia Sagastizábal
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Correspondence to Vincent Guigues.

Additional information

The first author’s research was supported by CNPq grant No. 382.851/07-4. The work of the second author was partially supported by grants CNPq 303840/2011-0, AFOSR FA9550-08-1-0370, NSF DMS 0707205, PRONEX-Optimization, and FAPERJ. Claudia Sagastizábal: On leave from INRIA Rocquencourt, France.

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Guigues, V., Sagastizábal, C. Risk-averse feasible policies for large-scale multistage stochastic linear programs. Math. Program. 138, 167–198 (2013). https://doi.org/10.1007/s10107-012-0592-1

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