# Apollo-MILP: An Alternating Prediction-Correction Neural Solving Framework for Mixed-Integer Linear Programming

arXiv: https://arxiv.org/abs/2503.01129 · HTML: https://mlsanity.com/paper/2503.01129/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 76 references that are in the map.
- First posted on arXiv 3 Mar 2025.

## Builds on (most cited first, 30 of 50)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon](https://mlsanity.com/paper/1811.06128/) (2018)
- [Exact Combinatorial Optimization with Graph Convolutional Neural Networks](https://mlsanity.com/paper/1906.01629/) (2019)
- [Solving Mixed Integer Programs Using Neural Networks](https://mlsanity.com/paper/2012.13349/) (2020)
- [Reinforcement Learning for Integer Programming: Learning to Cut](https://mlsanity.com/paper/1906.04859/) (2019)
- [Hybrid Models for Learning to Branch](https://mlsanity.com/paper/2006.15212/) (2020)
- [Parameterizing Branch-and-Bound Search Trees to Learn Branching Policies](https://mlsanity.com/paper/2002.05120/) (2020)
- [Learning To Cut By Looking Ahead: Cutting Plane Selection via Imitation Learning](https://mlsanity.com/paper/2206.13414/) (2022)
- [MIP-GNN: A Data-Driven Framework for Guiding Combinatorial Solvers](https://mlsanity.com/paper/2205.14210/) (2022)
- [Learning to Select Cuts for Efficient Mixed-Integer Programming](https://mlsanity.com/paper/2105.13645/) (2021)
- [Learning to Schedule Heuristics in Branch-and-Bound](https://mlsanity.com/paper/2103.10294/) (2021)
- [Learning to branch with Tree MDPs](https://mlsanity.com/paper/2205.11107/) (2022)
- [Learning Cut Selection for Mixed-Integer Linear Programming via Hierarchical Sequence Model](https://mlsanity.com/paper/2302.00244/) (2023)
- [Learning a Large Neighborhood Search Algorithm for Mixed Integer Programs](https://mlsanity.com/paper/2107.10201/) (2021)
- [A GNN-Guided Predict-and-Search Framework for Mixed-Integer Linear Programming](https://mlsanity.com/paper/2302.05636/) (2023)
- [The Machine Learning for Combinatorial Optimization Competition (ML4CO): Results and Insights](https://mlsanity.com/paper/2203.02433/) (2022)
- [On Representing Linear Programs by Graph Neural Networks](https://mlsanity.com/paper/2209.12288/) (2022)
- [Searching Large Neighborhoods for Integer Linear Programs with Contrastive Learning](https://mlsanity.com/paper/2302.01578/) (2023)
- [Learning Large Neighborhood Search Policy for Integer Programming](https://mlsanity.com/paper/2111.03466/) (2021)
- [Learning to Compare Nodes in Branch and Bound with Graph Neural Networks](https://mlsanity.com/paper/2210.16934/) (2022)
- [Lookback for Learning to Branch](https://mlsanity.com/paper/2206.14987/) (2022)
- [LMC: Fast Training of GNNs via Subgraph Sampling with Provable Convergence](https://mlsanity.com/paper/2302.00924/) (2023)
- [Machine Learning Augmented Branch and Bound for Mixed Integer Linear Programming](https://mlsanity.com/paper/2402.05501/) (2024)
- [Sample-Efficient Reinforcement Learning via Conservative Model-Based Actor-Critic](https://mlsanity.com/paper/2112.10504/) (2021)
- [On Representing Mixed-Integer Linear Programs by Graph Neural Networks](https://mlsanity.com/paper/2210.10759/) (2022)
- [A Deep Instance Generative Framework for MILP Solvers Under Limited Data Availability](https://mlsanity.com/paper/2310.02807/) (2023)
- [Structural Analysis of Branch-and-Cut and the Learnability of Gomory Mixed Integer Cuts](https://mlsanity.com/paper/2204.07312/) (2022)
- [Landscape Surrogate: Learning Decision Losses for Mathematical Optimization Under Partial Information](https://mlsanity.com/paper/2307.08964/) (2023)
- [MILP-StuDio: MILP Instance Generation via Block Structure Decomposition](https://mlsanity.com/paper/2410.22806/) (2024)
- [Learning Task-relevant Representations for Generalization via Characteristic Functions of Reward Sequence Distributions](https://mlsanity.com/paper/2205.10218/) (2022)

## Built on by (most cited first, 16 of 16)

- [Mixture-of-Experts Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2510.25803/) (2025)
- [SymMaP: Improving Computational Efficiency in Linear Solvers through Symbolic Preconditioning](https://mlsanity.com/paper/2510.24170/) (2025)
- [STNet: Spectral Transformation Network for Solving Operator Eigenvalue Problem](https://mlsanity.com/paper/2510.23986/) (2025)
- [RoME: Domain-Robust Mixture-of-Experts for MILP Solution Prediction across Domains](https://mlsanity.com/paper/2511.02331/) (2025)
- [Constraint Matters: Multi-Modal Representation for Reducing Mixed-Integer Linear programming](https://mlsanity.com/paper/2508.18742/) (2025)
- [Differentiable Integer Linear Programming is not Differentiable & it's not a mere technical problem](https://mlsanity.com/paper/2601.17800/) (2026)
- [A General Neural Backbone for Mixed-Integer Linear Optimization via Dual Attention](https://mlsanity.com/paper/2601.04509/) (2026)
- [FMIP: Joint Continuous-Integer Flow For Mixed-Integer Linear Programming](https://mlsanity.com/paper/2507.23390/) (2025)
- [Feature Augmentation of GNNs for ILPs: Local Uniqueness Suffices](https://mlsanity.com/paper/2509.21000/) (2025)
- [Designing faster mixed integer linear programming algorithm via learning the optimal path](https://mlsanity.com/paper/2601.16056/) (2026)
- [Learning to Solve Orienteering Problem with Time Windows and Variable Profits](https://mlsanity.com/paper/2603.06260/) (2026)
- [Towards Solving Polynomial-Objective Integer Programming with Hypergraph Neural Networks](https://mlsanity.com/paper/2603.19318/) (2026)
- [SRG: Score-based Relaxation-guided Generation for Mixed Integer Linear Programming](https://mlsanity.com/paper/2603.24033/) (2026)
- [Solving Integer Linear Programming with Parallel Tempering](https://mlsanity.com/paper/2605.29366/) (2026)
- [Autoresearch in Mixed-Integer Linear and Nonlinear Programming](https://mlsanity.com/paper/2609.39360/) (2026)
- [Reformulation-Contrastive Learning for Mixed Integer Programs](https://mlsanity.com/paper/2610.00730/) (2026)

## Abstract (arXiv)

Leveraging machine learning (ML) to predict an initial solution for mixed-integer linear programming (MILP) has gained considerable popularity in recent years. These methods predict a solution and fix a subset of variables to reduce the problem dimension. Then, they solve the reduced problem to obtain the final solutions. However, directly fixing variable values can lead to low-quality solutions or even infeasible reduced problems if the predicted solution is not accurate enough. To address this challenge, we propose an Alternating prediction-correction neural solving framework (Apollo-MILP) that can identify and select accurate and reliable predicted values to fix. In each iteration, Apollo-MILP conducts a prediction step for the unfixed variables, followed by a correction step to obtain an improved solution (called reference solution) through a trust-region search. By incorporating the predicted and reference solutions, we introduce a novel Uncertainty-based Error upper BOund (UEBO) to evaluate the uncertainty of the predicted values and fix those with high confidence. A notable feature of Apollo-MILP is the superior ability for problem reduction while preserving optimality, leading to high-quality final solutions. Experiments on commonly used benchmarks demonstrate that our proposed Apollo-MILP significantly outperforms other ML-based approaches in terms of solution quality, achieving over a 50% reduction in the solution gap.