Optimality of Affine Policies in Multi-stage Robust Optimization

TL;DR

This paper proves the optimality of disturbance-affine control policies in multi-stage robust optimization with convex costs, introducing a new proof method and fast algorithms for inventory management applications.

Contribution

It presents a novel proof technique establishing the optimality of affine policies and develops efficient algorithms for piecewise affine costs in robust control.

Findings

Affine policies are optimal in the specified robust optimization setting.

The new proof method links geometric properties to problem structure.

Fast algorithms are developed for inventory management applications.

Abstract

In this paper, we show the optimality of a certain class of disturbance-affine control policies in the context of one-dimensional, constrained, multi-stage robust optimization. Our results cover the finite horizon case, with minimax (worst-case) objective, and convex state costs plus linear control costs. We develop a new proof methodology, which explores the relationship between the geometrical properties of the feasible set of solutions and the structure of the objective function. Apart from providing an elegant and conceptually simple proof technique, the approach also entails very fast algorithms for the case of piecewise affine state costs, which we explore in connection with a classical inventory management application.