  We introduce a variant of Multicut Decomposition Algorithms (MuDA), 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 and Limited Memory Level 1 cut
selection strategies, initially introduced for respectively Stochastic Dual
Dynamic Programming (SDDP) and Dual Dynamic Programming (DDP). 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 SDDP combined with our class of cut selection strategies. We
compare the performance of MuDA, SDDP, and their variants with cut selection
(using Level 1 and Limited Memory Level 1) on several instances of a portfolio
problem and of an inventory problem. On these experiments, in general, SDDP is
quicker (i.e., satisfies the stopping criterion quicker) than MuDA and cut
selection allows us to decrease the computational bulk with Limited Memory
Level 1 being more efficient (sometimes much more) than Level 1.
