Modeling Design and Control Problems Involving Neural Network Surrogates

TL;DR

This paper explores methods to embed neural network surrogates into nonlinear optimization problems, proposing formulations that improve convergence and solution quality for applications in engineering and adversarial attack generation.

Contribution

It introduces two novel formulations for neural network-based optimization problems and proves their stationarity equivalence, enabling more effective solution approaches.

Findings

Formulations as mixed-integer and complementarity problems are effective.

State-of-the-art solvers can be applied to these formulations.

Applications include engine design, adversarial attacks, and oil flow optimization.

Abstract

We consider nonlinear optimization problems that involve surrogate models represented by neural networks. We demonstrate first how to directly embed neural network evaluation into optimization models, highlight a difficulty with this approach that can prevent convergence, and then characterize stationarity of such models. We then present two alternative formulations of these problems in the specific case of feedforward neural networks with ReLU activation: as a mixed-integer optimization problem and as a mathematical program with complementarity constraints. For the latter formulation we prove that stationarity at a point for this problem corresponds to stationarity of the embedded formulation. Each of these formulations may be solved with state-of-the-art optimization methods, and we show how to obtain good initial feasible solutions for these methods. We compare our formulations on three practical applications arising in the design and control of combustion engines, in the generation of adversarial attacks on classifier networks, and in the determination of optimal flows in an oil well network.