A Deep Instance Generative Framework for MILP Solvers Under Limited Data Availability

TL;DR

This paper tackles data scarcity in MILP research by introducing G2MILP, the first deep generative framework for MILP instances. MILP problems are represented as weighted bipartite graphs (constraints and variables connected by coefficients), and a masked variational autoencoder is used to iteratively corrupt and reconstruct graphs, balancing novelty with preservation of structure and computational hardness. The model decomposes the decoder into four modules (Bias, Degree, Logits, Weights) to predict both graph topology and numerical coefficients, trained with an ELBO objective that couples reconstruction and latent regularization. Through benchmarks on structural similarity, hardness preservation, and downstream value-prediction tasks, G2MILP demonstrates the ability to generate realistic MILP instances and to meaningfully improve solver-related tasks when real data are limited, offering a pathway to adversarial data generation and solver enrichment.

Abstract

In the past few years, there has been an explosive surge in the use of machine learning (ML) techniques to address combinatorial optimization (CO) problems, especially mixed-integer linear programs (MILPs). Despite the achievements, the limited availability of real-world instances often leads to sub-optimal decisions and biased solver assessments, which motivates a suite of synthetic MILP instance generation techniques. However, existing methods either rely heavily on expert-designed formulations or struggle to capture the rich features of real-world instances. To tackle this problem, we propose G2MILP, the first deep generative framework for MILP instances. Specifically, G2MILP represents MILP instances as bipartite graphs, and applies a masked variational autoencoder to iteratively corrupt and replace parts of the original graphs to generate new ones. The appealing feature of G2MILP is that it can learn to generate novel and realistic MILP instances without prior expert-designed formulations, while preserving the structures and computational hardness of real-world datasets, simultaneously. Thus the generated instances can facilitate downstream tasks for enhancing MILP solvers under limited data availability. We design a suite of benchmarks to evaluate the quality of the generated MILP instances. Experiments demonstrate that our method can produce instances that closely resemble real-world datasets in terms of both structures and computational hardness. The deliverables are released at https://miralab-ustc.github.io/L2O-G2MILP.

Figures (5)

• Figure 1: Overview of G2MILP. (a) Masking Process ${\tilde{p}}({\Tilde{{\mathcal{G}}}}|{\mathcal{G}})$. Given a MILP instance, which is represented as a bipartite graph ${\mathcal{G}}$, we randomly label a constraint vertex ${\tilde{v}}$ as [mask] to obtain the masked graph ${\Tilde{{\mathcal{G}}}}$. (b) Encoder $q_{\bm{\phi}}({\mathbf{Z}}|{\mathcal{G}})$. The encoder is $\text{GNN}_{{\bm{\phi}}}$ followed by two networks, ${\bm{\mu}}_{\bm{\phi}}$ and ${\bm{\Sigma}}_{\bm{\phi}}$, for resampling. During training, we use the encoder to obtain the latent vectors ${\bm{z}}_{v_i}$ and ${\bm{z}}_{w_j}$ for all vertices. (c) Decoder $p_{\bm{\theta}}({\mathcal{G}}|{\Tilde{{\mathcal{G}}}},{\mathbf{Z}})$. We use $\text{GNN}_{{\bm{\phi}}}$ to obtain the node features ${\bm{h}}_{{\tilde{v}}}$ and ${\bm{h}}_{w_j}$. Then four modules work cooperatively to reconstruct the original graph ${\mathcal{G}}$ based on the node features and the latent vectors. They sequentially determine ① the bias terms, ② the degrees, ③ the logits, and ④ the weights. During inference, the model is decoder-only, and we draw the latent vectors from a standard Guassian distribution to introduce randomness. We repeat the above mask-and-generate process several times so as to produce new instances.
• Figure 2: Results of the optimal value prediction task. Bars indicate the relative MSE to the model trained on the original training sets, and lines represent the relative performance improvement.
• Figure 3: The t-SNE visualization of MILP instance representations for MIK. Each point represents an instance. Red points are from the training set and blue points are instances generated by G2MILP.
• Figure 4: (a) Distributional similarity score (higher is better) and (b) Relative MSE (lower is better) v.s. masking ratio $\eta$.
• Figure 5: The t-SNE visualization of MILP instance representations for MIK. Each point represents an instance. Red points are from the training set, blue points are instances generated by G2MILP, and green points are instances generated by Random.