In this paper, we consider multistage robust convex optimization problems of the minimax type. We approximate the given robust problem by a sampled subproblem, where instead of looking for the worst case among the infinite and typically uncountable set of uncertain parameters, we consider only the worst case among a randomly selected subset of parameters. By adopting such a strategy, two main questions arise: (1) Can we quantify the error committed by the random approximation, especially as a function of the sample size? (2) If the sample size tends to infinity, does the optimal value converge to the ``true'' optimal value? Both questions will be answered in this paper. An explicit bound on the probability of violation is given and chain of lower bounds on the original multistage robust optimization problem provided. Numerical results dealing with a multistage inventory management problem show the efficiency of the proposed approach.
Submitted for publication in international journal on November 19th 2019.