Complexity of Stochastic Dual Dynamic Programming
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Stochastic dual dynamic programming is a cutting plane type algorithm for multi-stage stochastic optimization developed more than 30 years ago. In spite of its popularity in practice, there does not exist any performance guarantees on the convergence speed of this method. In this paper, we first establish the number of iteration, i.e., iteration complexity, required by a basic dynamic cutting plane method for solving relatively simple multi-stage optimization problems, by introducing novel mathematical tools including the saturation of search points. We then refine these basic tools and establish the iteration complexity for both deterministic and stochastic dual dynamic programming methods for solving more general multi-stage stochastic optimization problems under the standard stage-wise independence assumption. Our results indicate that the complexity of these methods mildly increases with respect to the number of stages T, in fact linearly dependent on T for discounted problems. Therefore, they are efficient for strategic decision making which involves a large number of stages, but with a relatively smaller number of decision variables in each stage. Without explicit discretization on the state and action spaces, these methods might also be pertinent to the related reinforcement learning and stochastic control areas.
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