Dynamic stochastic approximation for multi-stage stochastic optimization

In this paper, we consider multi-stage stochastic optimization problems with convex objectives and conic constraints at each stage. We present a new stochastic first-order method, namely the dynamic stochastic approximation (DSA) algorithm, for solving these types of stochastic optimization problems. We show that DSA can achieve an optimal O(1/epsilon 4) rate of convergence in terms of the total number of required scenarios when applied to a three-stage stochastic optimization problem. We further show that this rate of convergence can be improved to O(1/epsilon 2) when the objective function is strongly convex. We also discuss variants of DSA for solving more general multi-stage stochastic optimization problems with the number of stages T>3. The developed DSA algorithms only need to go through the scenario tree once in order to compute an epsilon-solution of the multi-stage stochastic optimization problem. As a result, the memory required by DSA only grows linearly with respect to the number of stages. To the best of our knowledge, this is the first time that stochastic approximation type methods are generalized for multi-stage stochastic optimization with T >= 3.

Automated summary

In this paper, a new dynamic stochastic approximation (DSA) algorithm is proposed for solving multi-stage stochastic optimization problems, achieving faster convergence under specific conditions. The algorithm only needs to traverse the scenario tree once to compute the solution of the multi-stage stochastic optimization problem, with memory requirements growing linearly with the number of stages.