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

TL;DR

Apollo-MILP targets the core challenge in MILP solving: how to safely reduce problem size by fixing variables inferred from predictions. By introducing an alternating prediction-correction loop and a novel UEBO metric, the method selects highly reliable fixed variables and iteratively reduces the MILP while preserving feasibility and (often) optimality. The correction step uses a trust-region search to produce a reference solution, and UEBO combines prediction uncertainty with prediction-correction discrepancy to guide fixes, backed by theoretical results on consistency and precision. Empirically, Apollo-MILP substantially improves solution quality across multiple benchmarks, achieving over 50% reduction in solution gap and often outperforming extended runtime baselines, including warm-started solvers, on real-world MILP instances.

Abstract

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.

Figures (5)

• Figure 1: Illustration of Neural Diving (ND) and Predict-and-Search (PS). For a given MILP problem, both methods begin by using a GNN predictor to generate an initial solution $\hat{{\bm{x}}}$ and construct a partial solution ${\hat{{\bm{x}}}[P]}$. ND then fixes the variable values in this partial solution and optimizes the reduced problem. While PS searches within a neighborhood around the partial solution.
• Figure 2: The overview of Apollo-MILP. Apollo-MILP operates through an iterative process that alternates between prediction and correction steps to reduce the original MILP problem progressively. In the prediction step, Apollo-MILP (1) employs a GNN to generate a partial solution. In the correction step, (2) a trust region-based search is conducted to refine this solution to obtain the reference solution. (3) The proposed variable fixing criterion, UEBO, is then calculated to identify which variables should be fixed. (4) Finally, we reduce the problem dimension by enforcing the selected variable values to fix values.
• Figure 3: The primal gap of the approaches as the solving process proceeds. Our methods are implemented using Gurobi, with a time limit set to 1,000s, and we average the results across 100 testing instances. A lower primal gap for our method indicates stronger convergence performance.
• Figure 4: The reproduced results of SCIP and ND+SCIP on the NNV dataset.
• Figure 5: The reproduced results of SCIP and ConPS+SCIP on the IP dataset.

Theorems & Definitions (6)

• Proposition 1: Uncertainty-Based Error Upper Bound
• Definition 1
• Theorem 2: UEBO and Consistency
• Theorem 3: Precision Improvement Guarantee
• Corollary 4
• Example 1