Approximation of multistage stochastic programming problems by smoothed quantization

Abstract

We present an approximation technique for solving multistage stochastic programming problems with an underlying Markov stochastic process. This process is approximated by a discrete skeleton process, which is consequently smoothed down by means of the original unconditional distribution. Approximated in this way, the problem is solvable by means of Markov Stochastic Dual Dynamic Programming. We state an upper bound for the nested distance between the exact process and its approximation and discuss its convergence in the one-dimensional case. We further propose an adjustment of the approximation, which guarantees that the approximate problem is bounded. Finally, we apply our technique to a real-life production-emission trading problem and demonstrate the performance of its approximation given the “true” distribution of the random parameters.