Tropical Dynamic Programming for Lipschitz Multistage Stochastic Programming

TL;DR

This paper introduces Tropical Dynamic Programming (TDP), an algorithm that approximates Bellman value functions in multistage stochastic programming using min-plus and max-plus linear combinations, with proven convergence.

Contribution

The paper presents TDP, a novel method for approximating value functions in MSP that converges asymptotically, extending the capabilities of existing approximation techniques.

Findings

TDP converges asymptotically to Bellman value functions.

TDP effectively handles Lipschitz MSP with linear dynamics.

The method is illustrated on MSP with linear dynamics and polyhedral costs.

Abstract

We present an algorithm called Tropical Dynamic Programming (TDP) which builds upper and lower approximations of the Bellman value functions in risk-neutral Multistage Stochastic Programming (MSP), with independent noises of finite supports. To tackle the curse of dimensionality, popular parametric variants of Approximate Dynamic Programming approximate the Bellman value function as linear combinations of basis functions. Here, Tropical Dynamic Programming builds upper (resp. lower) approximations of a given value function as min-plus linear (resp. max-plus linear) combinations of "basic functions". At each iteration, TDP adds a new basic function to the current combination following a deterministic criterion introduced by Baucke, Downward and Zackeri in 2018 for a variant of Stochastic Dual Dynamic Programming. We prove, for every Lipschitz MSP, the asymptotic convergence of the generated approximating functions of TDP to the Bellman value functions on sets of interest. We illustrate this result on MSP with linear dynamics and polyhedral costs.