# MIPaaL: Mixed Integer Program as a Layer

## TL;DR

This paper introduces a novel method for decision-focused learning that enables differentiation through Mixed Integer Linear Programs (MIPs) using a cutting planes approach, improving decision quality in complex optimization problems.

## Contribution

It extends decision-focused learning to general MIPs by developing a differentiation technique via cutting planes, supporting arbitrary linear constraints over discrete and continuous variables.

## Key findings

- Outperforms standard two-phase approaches in real-world tasks
- Effective differentiation through MIPs improves decision quality
- Supports broad class of problems with complex constraints

## Abstract

Machine learning components commonly appear in larger decision-making pipelines; however, the model training process typically focuses only on a loss that measures accuracy between predicted values and ground truth values. Decision-focused learning explicitly integrates the downstream decision problem when training the predictive model, in order to optimize the quality of decisions induced by the predictions. It has been successfully applied to several limited combinatorial problem classes, such as those that can be expressed as linear programs (LP), and submodular optimization. However, these previous applications have uniformly focused on problems from specific classes with simple constraints. Here, we enable decision-focused learning for the broad class of problems that can be encoded as a Mixed Integer Linear Program (MIP), hence supporting arbitrary linear constraints over discrete and continuous variables. We show how to differentiate through a MIP by employing a cutting planes solution approach, which is an exact algorithm that iteratively adds constraints to a continuous relaxation of the problem until an integral solution is found. We evaluate our new end-to-end approach on several real world domains and show that it outperforms the standard two phase approaches that treat prediction and prescription separately, as well as a baseline approach of simply applying decision-focused learning to the LP relaxation of the MIP.