Exact quantization of multistage stochastic linear problems

TL;DR

This paper demonstrates that multistage linear problems with any cost distribution can be exactly represented on a finite scenario tree, revealing new complexity insights and the polyhedral structure of cost-to-go functions.

Contribution

It establishes an exact quantization of multistage linear problems onto finite scenario trees, independent of cost distribution, and proves fixed-parameter tractability.

Findings

Expected cost-to-go functions are polyhedral and affine within chambers.

Multistage linear problems are fixed-parameter tractable.

Polyhedral structure is independent of cost distribution.

Abstract

We show that the multistage linear problem (MSLP) with an arbitrary cost distribution is equivalent to a MSLP on a finite scenario tree. We establish this exact quantization result by analyzing the polyhedral structure of MSLPs. In particular, we show that the expected cost-to-go functions are polyhedral and affine on the cells of a chamber complex, which is independent of the cost distribution. This leads to new complexity results, showing that MSLP is fixed-parameter tractable.