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Cut-sharing across trees and efficient sequential sampling for SDDP with uncertainty in the RHS

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  • Pedro Borges

    (Instituto de Matemática Pura e Aplicada)

Abstract

Multistage stochastic optimization problems (MSOP) are a commonly used paradigm to model many decision processes in energy and finance. Usually, a set of scenarios (the so-called tree) describing the stochasticity of the problem are obtained and the Stochastic Dual Dynamic Programming (SDDP) algorithm is often used to compute policies. Quite often, the uncertainty affects only the right-hand side (RHS) of the optimization problems in consideration. After solving a MSOP, one naturally wants to know if the solution obtained depends on the scenarios and by how much. In this paper we show that when a MSOP with stage-wise independent realizations has only RHS uncertainties, solving one tree using SDDP provides a valid lower bound for all trees with the same number of scenarios per stage without any additional computational effort. The only change to the traditional SDDP is the way cuts are calculated. Once the first tree is solved approximately, a computational assessment of the statistical significance of the current number of scenarios per stage is performed, solving for each new sampled tree, an easy LP to get a valid lower bound for the new tree. The objective of the paper is to estimate by how much the lower bound of the first tree depends on chance. The result of the computational assessment are fast estimates of the mean, variance and max variation of lower bounds across many trees. If the variance of the calculated lower bounds is small, we conclude that the cutting planes model has a small sensitivity to the trees sampled. Otherwise, we increase the number of scenarios per stage and repeat. We do not make assumptions on the distributions of the random variables. The results are not asymptotic. Our method has applications to the determination of the correct number of scenarios per stage. Extensions for uncertainties in the objective only are possible via the dual SDDP. We test our method numerically and verify the correctness of the cut-sharing technique.

Suggested Citation

  • Pedro Borges, 2022. "Cut-sharing across trees and efficient sequential sampling for SDDP with uncertainty in the RHS," Computational Optimization and Applications, Springer, vol. 82(3), pages 617-647, July.
    • Handle: RePEc:spr:coopap:v:82:y:2022:i:3:d:10.1007_s10589-022-00376-w
    DOI: 10.1007/s10589-022-00376-w
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    References listed on IDEAS

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