Abstract
We introduce a variant of Multicut Decomposition Algorithms, called CuSMuDA (Cut Selection for Multicut Decomposition Algorithms), for solving multistage stochastic linear programs that incorporates a class of cut selection strategies to choose the most relevant cuts of the approximate recourse functions. This class contains Level 1 (Philpott et al. in J Comput Appl Math 290:196–208, 2015) and Limited Memory Level 1 (Guigues in Eur J Oper Res 258:47–57, 2017) cut selection strategies, initially introduced for respectively Stochastic Dual Dynamic Programming and Dual Dynamic Programming. We prove the almost sure convergence of the method in a finite number of iterations and obtain as a by-product the almost sure convergence in a finite number of iterations of Stochastic Dual Dynamic Programming combined with our class of cut selection strategies. We compare the performance of Multicut Decomposition Algorithms, Stochastic Dual Dynamic Programming, and their variants with cut selection (using Level 1 and Limited Memory Level 1) on several instances of a portfolio problem. On these experiments, in general, Stochastic Dual Dynamic Programming is quicker (i.e., satisfies the stopping criterion quicker) than Multicut Decomposition Algorithms and cut selection allows us to decrease the computational bulk with Limited Memory Level 1 being more efficient (sometimes much more) than Level 1.
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Notes
We considered in particular instances where M is less than the number \(2n+1\) of constraints of the subproblems to give MuDA a chance (according to Birge et al. (1996), cases where MuDA may be competitive with (single cut) SDDP satisfy this requirement).
We see in particular that the bounds for all algorithms are very close to each other at the last iteration and that, as expected, the lower bounds increase and the upper bounds tend to decrease along iterations. If we do not know the optimal value \({\mathcal {Q}}_{1}( x_{0})\), these observations (that we checked on all instances) are a good indication that all algorithms were correctly implemented.
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Acknowledgements
The second author’s research was partially supported by an FGV Grant, CNPq Grant 401371/2014-0 and FAPERJ grant E-26/201.599/2014. The authors wish to thank Vincent Leclère for helpful discussions.
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Bandarra, M., Guigues, V. Single cut and multicut stochastic dual dynamic programming with cut selection for multistage stochastic linear programs: convergence proof and numerical experiments. Comput Manag Sci 18, 125–148 (2021). https://doi.org/10.1007/s10287-021-00387-8
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DOI: https://doi.org/10.1007/s10287-021-00387-8
Keywords
- Stochastic programming
- Stochastic dual dynamic programming
- Multicut decomposition algorithm
- Portfolio selection