Time Block Decomposition of Multistage Stochastic Optimization Problems

Multistage stochastic optimization problems are, by essence, complex as their solutions are functions of both stages and uncertainties. Their large scale nature makes decomposition methods appealing, like dynamic programming which is a sequential decomposition using a state variable defined at all stages. By contrast, in this paper we introduce the notion of state reduction by time blocks, that is, at stages that are not necessarily all the original stages. Then, we prove a dynamic programming equation with value functions that are functions of a state only at some stages. This equation crosses over time blocks, but involves a dynamic optimization inside each block. We illustrate our contribution by showing its potential in three applications in multistage stochastic optimization: mixing dynamic programming and stochastic programming, two-timescale optimization problems, decision-hazard-decision optimization problems.