9 March 2026 |

From perception to constrained decisions

Embedding optimization in neural networks

Georg Martius’ project, “Embedding combinatorial reasoning into learning control policies” addressed a practical gap in AI's basic machinery: connecting data-driven neural networks with decision-making so they can adhere to clear structures, rules, and constraints. The project developed foundations for integrating combinatorial optimization into learning systems. It has potential applications wherever planning and resource allocation are central, such as in robotics and logistics. The project was funded by the Cyber Valley Research Fund and was carried out between 2021 and 2024.

Martius’ project addressed a key limitation of today’s machine-learning systems. While neural networks can learn powerful representations from data (for example, from images or sensor signals), many real-world decisions must also follow strict rules, such as budgets, capacities, safety limits, or “must/ must not” constraints. These rule-based decisions are often handled by specialized optimization software, but such tools are traditionally difficult to integrate into learning-based systems.

Martius’ research created methods that make it easier to combine neural networks with combinatorial optimization components. This enables systems to learn from data while also drawing on proven optimization techniques to make decisions that respect constraints. Rather than replacing classical solvers, the project developed ways to use them as trainable building blocks inside larger learning pipelines.

The project also explored “neuro-algorithmic” approaches to decision-making and control, investigating how structured algorithms can improve generalization to new problems compared with purely data-driven approaches.

Overall, the project provides a foundation for building AI systems that connect perception with constraint-aware decision modules. This is an important step toward robust solutions in areas such as robotics, planning, logistics, and resource allocation.

This project produced the following peer-reviewed publications:

Anselm Paulus, Michal Rolínek, Vít Musil, Brandon Amos, and Georg Martius. “CombOptNet: Fit the Right NP-Hard Problem by Learning Integer Programming Constraints”. In: Proceedings of the 38th International Conference on Machine Learning. Vol. 139. Proceedings of Machine Learning Research. PMLR, July 2021, pp. 8443–8453. url: https://proceedings.mlr.press/v139/paulus21a.html.

Marin Vlastelica, Michal Rolinek, and Georg Martius. “Neuro-algorithmic Policies Enable Fast Combinatorial Generalization”. In: Proceedings of the 2021 International Conference on Machine Learning (ICML). July 2021. url: https://proceedings.mlr.press/v139/vlastelica21a.html.

Marco Bagatella, Mirek Olšák, Michal Rolínek, and Georg Martius. “Planning from Pixels in Environments with Combinatorially Hard Search Spaces”. In: Advances in Information Processing Systems (NeurIPS 2021). 2021. url: https://openreview.net/forum?id=XgGUUaKgips.

Subham Sahoo, Anselm Paulus, Marin Vlastelica, Vít Musil, Volodymyr Kuleshov, and Georg Martius. “Backpropagation through Combinatorial Algorithms: Identity with Projection Works”. In: Proceedings of the Eleventh International Conference on Learning Representations. May 2023. url: https://openreview.net/forum?id=JZMR727O29.

A. Paulus, G. Martius, and V. Musil. “LPGD: A General Framework for Backpropagation through Embedded Optimization Layers”. In: Proceedings of the 41st International Conference on Machine Learning (ICML). Vol. 235. Proceedings of Machine Learning Research. PMLR, July 2024, pp. 39989–40014. url: https://proceedings.mlr.press/v235/paulus24a.html.

Georg Martius is a Professor in the Department of Computer Science at the University of Tübingen.

About the Cyber Valley Research Fund

The Cyber Valley Research Fund was established to support Cyber Valley research groups undertake basic research in the fields of artificial intelligence and robotics. The fund totaled five million euros, including contributions from six of Cyber Valley’s founding corporate partners: Amazon, BMW, Bosch, IAV, Mercedes-Benz, Porsche, and ZF. It supported 20 research projects, the first of which began in 2020, and the final of which will conclude in 2026.