# Input convex neural networks as surrogates in mathematical optimisation

arXiv: https://arxiv.org/abs/2608.09707 · HTML: https://mlsanity.com/paper/2608.09707/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 47 references that are in the map.
- First posted on arXiv 10 Aug 2026.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [A Survey of Model Compression and Acceleration for Deep Neural Networks](https://mlsanity.com/paper/1710.09282/) (2017)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Bounding and Counting Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1711.02114/) (2017)
- [Strong mixed-integer programming formulations for trained neural networks](https://mlsanity.com/paper/1811.08359/) (2018)
- [Maximum Resilience of Artificial Neural Networks](https://mlsanity.com/paper/1705.01040/) (2017)
- [Parallelizing the dual revised simplex method](https://mlsanity.com/paper/1503.01889/) (2015)
- [When Deep Learning Meets Polyhedral Theory: A Survey](https://mlsanity.com/paper/2305.00241/) (2023)
- [JuMP 1.0: Recent improvements to a modeling language for mathematical optimization](https://mlsanity.com/paper/2206.03866/) (2022)
- [ReLU Networks as Surrogate Models in Mixed-Integer Linear Programs](https://mlsanity.com/paper/1907.03140/) (2019)
- [Mixed-Integer Optimization with Constraint Learning](https://mlsanity.com/paper/2111.04469/) (2021)
- [Physics Informed Neural Networks for Control Oriented Thermal Modeling of Buildings](https://mlsanity.com/paper/2111.12066/) (2021)
- [Input Convex Neural Networks for Building MPC](https://mlsanity.com/paper/2011.13227/) (2020)
- [Optimization Over Trained Neural Networks: Taking a Relaxing Walk](https://mlsanity.com/paper/2401.03451/) (2024)
- [P-split formulations: A class of intermediate formulations between big-M and convex hull for disjunctive constraints](https://mlsanity.com/paper/2202.05198/) (2022)
- [Learning Optimal Power Flow Value Functions with Input-Convex Neural Networks](https://mlsanity.com/paper/2310.04605/) (2023)
- [Convexity in ReLU Neural Networks: beyond ICNNs?](https://mlsanity.com/paper/2501.03017/) (2025)
- [Fast Explicit Machine Learning-Based Model Predictive Control of Nonlinear Processes Using Input Convex Neural Networks](https://mlsanity.com/paper/2408.06580/) (2024)
- [ICNN-enhanced 2SP: Leveraging input convex neural networks for solving two-stage stochastic programming](https://mlsanity.com/paper/2505.05261/) (2025)
- [MathOptAI.jl: Embed trained machine learning predictors into JuMP models](https://mlsanity.com/paper/2507.03159/) (2025)

## Abstract (arXiv)

Embedding trained neural networks as surrogates within optimisation problems is an established practice in operations research. The prevailing approach uses feedforward neural networks (FNNs) with ReLU activations, whose piecewise-linear structure admits an exact but computationally intensive mixed-integer programming (MIP) reformulation as the networks grow. We advocate input convex neural networks (ICNNs) as structurally superior surrogates when the underlying response is approximately convex or concave. The convex architecture offers two computational advantages. First, the ICNN-MIP formulation tends to yield a tighter linear programming (LP) relaxation than its FNN-MIP counterpart, with no integrality gap in favourable instances. Second, ICNNs uniquely admit an LP-based reformulation via epigraph representations of ReLU activations, though this embedding is not always exact. When it is not, we exploit the properties of ICNNs to construct the strongest continuous relaxation over box domains, namely, the convex hull of the ICNN's graph, bounded below by the epigraph and above by the concave envelope; this construction is tractable under input convexity but hard for general ReLU networks. On this basis, we develop a branch-and-bound algorithm that builds this relaxation at each node, branches directly on input variables rather than intermediate variables as in MIP reformulations, and terminates at the root node whenever the epigraph embedding is valid. Case studies on humanitarian food aid, oil well routing, and wine blending show that ICNN surrogates match FNN accuracy and deliver gains in solve time and scalability, supporting ICNN as the default surrogate when the underlying function is convex, concave, or well-approximated as such.