A Unified Framework for Multistage and Multilevel Mixed Integer Linear Optimization

TL;DR

This paper presents a unified mathematical framework for multistage and multilevel mixed integer linear optimization problems, enabling shared solution strategies and insights into their structure, especially focusing on the two-stage case and solution techniques.

Contribution

It introduces a common framework for multilevel and multistage mixed integer linear problems, connecting their structures and proposing new solution methods like Benders-like decomposition and cutting planes.

Findings

Unified framework reveals common structure of the two problem types.

Analysis of the value function and duality in mixed integer problems.

Development of Benders-like and cutting plane solution techniques.

Abstract

We introduce a unified framework for the study of multilevel mixed integer linear optimization problems and multistage stochastic mixed integer linear optimization problems with recourse. The framework highlights the common mathematical structure of the two problems and allows for the development of a common algorithmic framework. Focusing on the two-stage case, we investigate, in particular, the nature of the value function of the second-stage problem, highlighting its connection to dual functions and the theory of duality for mixed integer linear optimization problems, and summarize different reformulations. We then present two main solution techniques, one based on a Benders-like decomposition to approximate either the risk function or the value function, and the other one based on cutting plane generation.