# Interior Point Solving for LP-based prediction+optimisation

arXiv: https://arxiv.org/abs/2010.13943 · HTML: https://mlsanity.com/paper/2010.13943/

## Key facts

- Cited by 83 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Oct 2020.

## Built on by (most cited first, 30 of 83)

- [Combinatorial optimization and reasoning with graph neural networks](https://mlsanity.com/paper/2102.09544/) (2021)
- [End-to-End Constrained Optimization Learning: A Survey](https://mlsanity.com/paper/2103.16378/) (2021)
- [Decision-Focused Learning: Foundations, State of the Art, Benchmark and Future Opportunities](https://mlsanity.com/paper/2307.13565/) (2023)
- [Implicit MLE: Backpropagating Through Discrete Exponential Family Distributions](https://mlsanity.com/paper/2106.01798/) (2021)
- [A Survey of Contextual Optimization Methods for Decision Making under Uncertainty](https://mlsanity.com/paper/2306.10374/) (2023)
- [Decision-Focused Learning: Through the Lens of Learning to Rank](https://mlsanity.com/paper/2112.03609/) (2021)
- [PyEPO: A PyTorch-based End-to-End Predict-then-Optimize Library for Linear and Integer Programming](https://mlsanity.com/paper/2206.14234/) (2022)
- [Backpropagation through Combinatorial Algorithms: Identity with Projection Works](https://mlsanity.com/paper/2205.15213/) (2022)
- [Learning with Combinatorial Optimization Layers: a Probabilistic Approach](https://mlsanity.com/paper/2207.13513/) (2022)
- [Integrated Conditional Estimation-Optimization](https://mlsanity.com/paper/2110.12351/) (2021)
- [Alternating Differentiation for Optimization Layers](https://mlsanity.com/paper/2210.01802/) (2022)
- [Efficient differentiable quadratic programming layers: an ADMM approach](https://mlsanity.com/paper/2112.07464/) (2021)
- [End-to-End Stochastic Optimization with Energy-Based Model](https://mlsanity.com/paper/2211.13837/) (2022)
- [Gen-DFL: Decision-Focused Generative Learning for Robust Decision Making](https://mlsanity.com/paper/2502.05468/) (2025)
- [Backpropagation of Unrolled Solvers with Folded Optimization](https://mlsanity.com/paper/2301.12047/) (2023)
- [Decision-Focused Learning without Differentiable Optimization: Learning Locally Optimized Decision Losses](https://mlsanity.com/paper/2203.16067/) (2022)
- [Adaptive Perturbation-Based Gradient Estimation for Discrete Latent Variable Models](https://mlsanity.com/paper/2209.04862/) (2022)
- [Neuro-Symbolic Constraint Programming for Structured Prediction](https://mlsanity.com/paper/2103.17232/) (2021)
- [Predict+Optimize for Packing and Covering LPs with Unknown Parameters in Constraints](https://mlsanity.com/paper/2209.03668/) (2022)
- [Differentiating Through Integer Linear Programs with Quadratic Regularization and Davis-Yin Splitting](https://mlsanity.com/paper/2301.13395/) (2023)
- [Score Function Gradient Estimation to Widen the Applicability of Decision-Focused Learning](https://mlsanity.com/paper/2307.05213/) (2023)
- [End-to-End Cost-Effective Incentive Recommendation under Budget Constraint with Uplift Modeling](https://mlsanity.com/paper/2408.11623/) (2024)
- [Decision Focused Causal Learning for Direct Counterfactual Marketing Optimization](https://mlsanity.com/paper/2407.13664/) (2024)
- [Solver-Free Decision-Focused Learning for Linear Optimization Problems](https://mlsanity.com/paper/2505.22224/) (2025)
- [Benchmarking PtO and PnO Methods in the Predictive Combinatorial Optimization Regime](https://mlsanity.com/paper/2311.07633/) (2023)
- [CaVE: A Cone-Aligned Approach for Fast Predict-then-optimize with Binary Linear Programs](https://mlsanity.com/paper/2312.07718/) (2023)
- [Differentiable Distributionally Robust Optimization Layers](https://mlsanity.com/paper/2406.16571/) (2024)
- [LPGD: A General Framework for Backpropagation through Embedded Optimization Layers](https://mlsanity.com/paper/2407.05920/) (2024)
- [Differentiable Nonlinear Model Predictive Control](https://mlsanity.com/paper/2505.01353/) (2025)
- [Feasibility-Aware Decision-Focused Learning for Predicting Parameters in the Constraints](https://mlsanity.com/paper/2510.04951/) (2025)

## Abstract (arXiv)

Solving optimization problems is the key to decision making in many real-life analytics applications. However, the coefficients of the optimization problems are often uncertain and dependent on external factors, such as future demand or energy or stock prices. Machine learning (ML) models, especially neural networks, are increasingly being used to estimate these coefficients in a data-driven way. Hence, end-to-end predict-and-optimize approaches, which consider how effective the predicted values are to solve the optimization problem, have received increasing attention. In case of integer linear programming problems, a popular approach to overcome their non-differentiabilty is to add a quadratic penalty term to the continuous relaxation, such that results from differentiating over quadratic programs can be used. Instead we investigate the use of the more principled logarithmic barrier term, as widely used in interior point solvers for linear programming. Specifically, instead of differentiating the KKT conditions, we consider the homogeneous self-dual formulation of the LP and we show the relation between the interior point step direction and corresponding gradients needed for learning. Finally our empirical experiments demonstrate our approach performs as good as if not better than the state-of-the-art QPTL (Quadratic Programming task loss) formulation of Wilder et al. and SPO approach of Elmachtoub and Grigas.