Learning optimal objective values for MILP

TL;DR

This work proposes a methodology for predicting the optimal objective value, or, equivalently, predicting if the current incumbent is optimal, using a predictor based on a graph neural network (GNN) architecture, together with a set of dynamic features.

Abstract

Modern Mixed Integer Linear Programming (MILP) solvers use the Branch-and-Bound algorithm together with a plethora of auxiliary components that speed up the search. In recent years, there has been an explosive development in the use of machine learning for enhancing and supporting these algorithmic components. Within this line, we propose a methodology for predicting the optimal objective value, or, equivalently, predicting if the current incumbent is optimal. For this task, we introduce a predictor based on a graph neural network (GNN) architecture, together with a set of dynamic features. Experimental results on diverse benchmarks demonstrate the efficacy of our approach, achieving high accuracy in the prediction task and outperforming existing methods. These findings suggest new opportunities for integrating ML-driven predictions into MILP solvers, enabling smarter decision-making and improved performance.

Figures (4)

• Figure 1: Optimal objective value prediction task. The MILP representation is computed after the root node has been processed. This serves as an input to a GNN that outputs a prediction $\tilde{z}^*$ of the optimal objective value.
• Figure 2: Phase analysis of three instance types. We divide the solution process into (1) Feasibility, in dark yellow, (2a) Improvement up to 5% to optimality, in light yellow, (2b) Improvement from 5% to optimal, in light purple, and (3) Proving, in dark purple. We also indicate when the first branching occurs. The data is averaged over 100 instances with 3 randomization seeds (i.e., 300 samples).
• Figure 3: Prediction accuracy of the different classifier models. We show the fraction of correctly classified samples (correct, in purple), the fraction of false positives (fp, dark yellow) and the fraction of false negatives (fn, light yellow).
• Figure 4: Feature importance of the dynamic models trained to predict phase transition for each of the benchmarks.