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 strategies to select the most relevant cuts of the approximate recourse functions. We prove the convergence of the method in a finite number of iterations and use it to solve six portfolio problems with direct transaction costs under return uncertainty and six inventory management problems under demand uncertainty. On all problem instances CuSMuDA is much quicker than MuDA: between 5.1 and 12.6 times quicker for the porfolio problems considered and between 6.4 and 15.7 times quicker for the inventory problems.
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