Provably Data-driven Lagrangian Relaxation for Mixed Integer Linear Programming

2026-05-18

AI Summary

This work addresses the data-driven learning of Lagrange multipliers in mixed-integer linear programming. By formulating multiplier learning as a statistical learning problem, it establishes the first minimax bounds on generalization error: an upper bound of $O(s^{1.5}/\sqrt{N})$ and a matching lower bound of $\Omega(s/\sqrt{N})$. The authors propose an averaged stochastic gradient ascent algorithm that achieves the optimal convergence rate of $\Theta(s/\sqrt{N})$. Furthermore, by incorporating a warm-start mechanism, they improve the convergence rate to $\Theta(s/N)$ under this setting. This study provides the first theoretical guarantees and a provably optimal learning framework for data-driven Lagrangian relaxation.

Abstract

Lagrangian Relaxation (LR) is a powerful technique for solving large-scale Mixed Integer Linear Programming (MILP), particularly those with decomposable structures, such as vehicle routing or unit commitment problems. By relaxing the coupling constraints, LR enables parallel subproblem solving and often yields tighter dual bounds than standard linear programming relaxations, which is crucial for efficient branch-and-bound pruning. While recent empirical work has shown promising results using machine learning to predict these multipliers, a theoretical understanding of such methods remains an open question. In this work, we bridge this gap by analyzing the problem of learning LR through the lens of Data-driven Algorithm Design, i.e., a statistical learning problem over a distribution of problem instances. Our contributions are as follows: first, we derive a generalization bound of $\mathcal{O}(s^{1.5}/\sqrt{N})$ for the learned multipliers, where $s$ is the number of coupling constraints and $N$ is the sample size. Second, we provide a minimax lower-bound of $Ω(s/\sqrt{N})$, proving that a linear dependency is unavoidable. Third, we constructively close this theoretical gap by proving that Stochastic Gradient Ascent (SGA) with averaging achieves the minimax optimal rate $Θ(s/\sqrt{N})$. Finally, we extend our framework to the learning-to-warm-start setting, proving that it achieves a fast, minimax-optimal rate of $Θ(s/N)$ and establishing a theoretical advantage over direct multiplier prediction.

Problem

• Lagrangian Relaxation
• Mixed Integer Linear Programming
• Data-driven Algorithm Design
• Generalization Bound
• Minimax Optimality

Innovation

• Lagrangian Relaxation
• Data-driven Algorithm Design
• Generalization Bound
• Minimax Optimality
• Learning to Warm Start