# ReLU Networks as Surrogate Models in Mixed-Integer Linear Programs

## TL;DR

This paper explores embedding ReLU neural networks as surrogate models in MILP problems, focusing on bound tightening techniques to improve computational efficiency and demonstrating their effectiveness with small networks.

## Contribution

It introduces bound tightening procedures that leverage input and output bounds to enhance MILP formulations with ReLU networks, improving solution times.

## Key findings

- Bound tightening reduces solution times significantly.
- Small ReLU networks are effective as surrogate models.
- Bound tightening improves MILP formulation efficiency.

## Abstract

We consider the embedding of piecewise-linear deep neural networks (ReLU networks) as surrogate models in mixed-integer linear programming (MILP) problems. A MILP formulation of ReLU networks has recently been applied by many authors to probe for various model properties subject to input bounds. The formulation is obtained by programming each ReLU operator with a binary variable and applying the big-M method. The efficiency of the formulation hinges on the tightness of the bounds defined by the big-M values. When ReLU networks are embedded in a larger optimization problem, the presence of output bounds can be exploited in bound tightening. To this end, we devise and study several bound tightening procedures that consider both input and output bounds. Our numerical results show that bound tightening may reduce solution times considerably, and that small-sized ReLU networks are suitable as surrogate models in mixed-integer linear programs.