# A Unified Framework for Multistage and Multilevel Mixed Integer Linear Optimization

arXiv: https://arxiv.org/abs/2104.09003 · HTML: https://mlsanity.com/paper/2104.09003/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Apr 2021.

## Built on by (most cited first, 10 of 10)

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- [A Primal-Dual Approach to Bilevel Optimization with Multiple Inner Minima](https://mlsanity.com/paper/2203.01123/) (2022)
- [Inexact bilevel stochastic gradient methods for constrained and unconstrained lower-level problems](https://mlsanity.com/paper/2110.00604/) (2021)
- [Convex Bi-Level Optimization Problems with Non-smooth Outer Objective Function](https://mlsanity.com/paper/2307.08245/) (2023)
- [An adaptively inexact first-order method for bilevel optimization with application to hyperparameter learning](https://mlsanity.com/paper/2308.10098/) (2023)
- [An Adaptively Inexact Method for Bilevel Learning Using Primal-Dual Style Differentiation](https://mlsanity.com/paper/2412.06436/) (2024)
- [Active Inverse Learning in Stackelberg Trajectory Games](https://mlsanity.com/paper/2308.08017/) (2023)
- [Decision-focused predictions via pessimistic bilevel optimization: complexity and algorithms](https://mlsanity.com/paper/2312.17640/) (2023)
- [On Stability in Optimistic Bilevel Optimization](https://mlsanity.com/paper/2408.13323/) (2024)
- [Soft-ECM: An extension of Evidential C-Means for complex data](https://mlsanity.com/paper/2507.13417/) (2025)

## Abstract (arXiv)

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.