# CaVE: A Cone-Aligned Approach for Fast Predict-then-optimize with Binary Linear Programs

arXiv: https://arxiv.org/abs/2312.07718 · HTML: https://mlsanity.com/paper/2312.07718/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 30 references that are in the map.
- First posted on arXiv 12 Dec 2023.

## Builds on (most cited first, 20 of 20)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017)
- [Differentiation of Blackbox Combinatorial Solvers](https://mlsanity.com/paper/1912.02175/) (2019)
- [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)
- [MIPaaL: Mixed Integer Program as a Layer](https://mlsanity.com/paper/1907.05912/) (2019)
- [Smart Predict-and-Optimize for Hard Combinatorial Optimization Problems](https://mlsanity.com/paper/1911.10092/) (2019)
- [Interior Point Solving for LP-based prediction+optimisation](https://mlsanity.com/paper/2010.13943/) (2020)
- [A Survey of Contextual Optimization Methods for Decision Making under Uncertainty](https://mlsanity.com/paper/2306.10374/) (2023)
- [Learning with Differentiable Perturbed Optimizers](https://mlsanity.com/paper/2002.08676/) (2020)
- [Decision-Focused Learning: Through the Lens of Learning to Rank](https://mlsanity.com/paper/2112.03609/) (2021)
- [Decision Trees for Decision-Making under the Predict-then-Optimize Framework](https://mlsanity.com/paper/2003.00360/) (2020)
- [Contrastive Losses and Solution Caching for Predict-and-Optimize](https://mlsanity.com/paper/2011.05354/) (2020)
- [Learning for Constrained Optimization: Identifying Optimal Active Constraint Sets](https://mlsanity.com/paper/1802.09639/) (2018)
- [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)
- [SurCo: Learning Linear Surrogates For Combinatorial Nonlinear Optimization Problems](https://mlsanity.com/paper/2210.12547/) (2022)
- [Leaving the Nest: Going Beyond Local Loss Functions for Predict-Then-Optimize](https://mlsanity.com/paper/2305.16830/) (2023)

## Built on by (most cited first, 8 of 8)

- [Solver-Free Decision-Focused Learning for Linear Optimization Problems](https://mlsanity.com/paper/2505.22224/) (2025)
- [Learning to Optimize for Mixed-Integer Non-linear Programming with Feasibility Guarantees](https://mlsanity.com/paper/2410.11061/) (2024)
- [Feasibility-Aware Decision-Focused Learning for Predicting Parameters in the Constraints](https://mlsanity.com/paper/2510.04951/) (2025)
- [A Solver-Free Training Method for Predict-then-Optimize](https://mlsanity.com/paper/2606.19587/) (2026)
- [Minimizing Surrogate Losses for Decision-Focused Learning using Differentiable Optimization](https://mlsanity.com/paper/2508.11365/) (2025)
- [Decision-Focused Federated Learning Under Heterogeneous Objectives and Constraints](https://mlsanity.com/paper/2604.20031/) (2026)
- [Scalable Decision-Focused Learning through Cost-Sensitive Regression](https://mlsanity.com/paper/2605.18005/) (2026)
- [Scaling Decision-Focused Learning to Large Problems with Lagrangian Decomposition](https://mlsanity.com/paper/2606.08797/) (2026)

## Abstract (arXiv)

The end-to-end predict-then-optimize framework, also known as decision-focused learning, has gained popularity for its ability to integrate optimization into the training procedure of machine learning models that predict the unknown cost (objective function) coefficients of optimization problems from contextual instance information. Naturally, most of the problems of interest in this space can be cast as integer linear programs. In this work, we focus on binary linear programs (BLPs) and propose a new end-to-end training method to predict-then-optimize. Our method, Cone-aligned Vector Estimation (CaVE), aligns the predicted cost vectors with the normal cone corresponding to the true optimal solution of a training instance. When the predicted cost vector lies inside the cone, the optimal solution to the linear relaxation of the binary problem is optimal. This alignment not only produces decision-aware learning models but also dramatically reduces training time as it circumvents the need to solve BLPs to compute a loss function with its gradients. Experiments across multiple datasets show that our method exhibits a favorable trade-off between training time and solution quality, particularly with large-scale optimization problems such as vehicle routing, a hard BLP that has yet to benefit from predict-then-optimize methods in the literature due to its difficulty.