Time Blocks Decomposition of Multistage Stochastic Optimization Problems

TL;DR

This paper introduces a novel time blocks decomposition method for multistage stochastic optimization, enabling reduced dynamic programming equations and potential applications in problems with multiple time scales.

Contribution

It proposes a new state reduction technique by time blocks and establishes a corresponding reduced dynamic programming framework, expanding the tools for complex stochastic problems.

Findings

Reduced dynamic programming equations derived for time blocks.

Potential for application in problems with multiple time scales.

Positioning within existing mathematical frameworks clarified.

Abstract

Multistage stochastic optimization problems are, by essence, complex as their solutions are indexed both by stages and by 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. 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 reduced dynamic programming equation. We position our result with respect to the most well-known mathematical frameworks for dynamic programming. We illustrate our contribution by showing its potential for applied problems with two time scales.