Single cut and multicut stochastic dual dynamic programming with cut selection for multistage stochastic linear programs: convergence proof and numerical experiments

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.