(392ak) Energy-Based Learning for Large-Scale Mixed-Integer Optimization

Mixed-Integer Linear Programming (MILP) is a powerful modeling paradigm for addressing complex combinatorial optimization problems across engineering, management, and supply chain planning. However, solving large-scale MILPs remains computationally challenging due to their combinatorial complexity and NP-hard nature. Machine learning (ML) methods have been explored to enhance or replace traditional optimization solvers by learning problem structure and guiding decisions [1–3]. While prior work has focused on learning heuristics or solution approximations [4–5], such approaches often predict directly over discrete variables, leading to infeasible or suboptimal outcomes due to their inability to explicitly incorporate feasibility constraints and domain-specific knowledge.

In this work, we introduce an energy-based learning (EBL) framework to enhance MILP solving by leveraging deep learning for structure-aware and optimization-compatible inference. Our method builds on our previous deep learning enhanced MIP pipeline, where artificial neural networks (ANNs) and convolutional neural networks (CNNs) were trained to approximate complicating binary variables in large-scale supply chain optimization problems [5]. However, direct prediction of binary decisions often leads to infeasibility or suboptimality, especially when the underlying constraints are complex or tightly coupled.

To address this, we reframe the problem using energy-based learning [6], where the trained neural networks define an energy function over the decision space [7]. Instead of predicting decisions directly, we infer optimal solutions by minimizing the learned energy function subject to MIP constraints, effectively integrating learning and optimization. The energy function is trained on precomputed MILP instances using contrastive learning to assign low energy to feasible and optimal solutions, and high energy to infeasible or poor-performing ones. This enables the model to generalize beyond training data and preserve feasibility during inference.

We apply our framework to a flow-based MILP model for investment and tactical planning in personalized medicine supply chains [8], demonstrating that energy-based inference consistently improves solution quality and feasibility compared to direct prediction. Furthermore, by embedding domain-specific constraints within the learned energy model, our method leverages both data-driven insights and structured problem knowledge, bridging the gap between black-box ML methods and interpretable optimization models.

References

[1] Y. Bengio, A. Lodi, A. Prouvost, "Machine learning for combinatorial optimization: A methodological tour d’horizon," European Journal of Operational Research, 290(2):405–421, 2021.

[2] M. Gasse, D. Chételat, N. Ferroni, L. Charlin, A. Lodi, "Exact combinatorial optimization with graph convolutional neural networks," Advances in neural information processing systems (NeurIPS), 32, 2019.

[3] I. Mitrai, P. Daoutidis, "Accelerating process control and optimization via machine learning: A review," arXiv preprint, arXiv:2412.18529, 2024.

[4] B. Abbasi, T. Babaei, Z. Hosseinifard, K. Smith-Miles, M. Dehghani, "Predicting solutions of large-scale optimization problems via machine learning: A case study in blood supply chain management," Computers & Operations Research, 119:104941, 2020.

[5] N. Triantafyllou, M.M. Papathanasiou, "Deep learning enhanced mixed integer optimization: Learning to reduce model dimensionality," Computers & Chemical Engineering, 187:108725, 2024.

[6] Y. LeCun, S. Chopra, R. Hadsell, F.J. Huang, A. Ranzato, "A tutorial on energy-based learning," in Predicting Structured Data, MIT Press, 2006.

[7] V. Nair, S. Bartunov, F. Gimeno, I. von Glehn, P. Lichocki, I. Lobov, B. O’Donoghue, N. Sonnerat, C. Tjandraatmadja, P. Wang, R. Addanki, T. Hapuarachchi, T. Keck, J. Keeling, P. Kohli, I. Ktena, Y. Li, O. Vinyals, Y. Zwols, "Solving mixed integer programs using neural networks," arXiv preprint, arXiv:2012.13349, 2020.

[8] N. Triantafyllou, A. Bernardi, M. Lakelin, N. Shah, M.M. Papathanasiou, "A digital platform for the design of patient-centric supply chains," Scientific Reports, 12:17365, 2022.