Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Data-driven Mixed Integer Optimization through Probabilistic Multi-variable Branching

Yanguang Chen, Wenzhi Gao, Wanyu Zhang, Dongdong Ge, Huikang Liu, Yinyu Ye

TL;DR

This work presents PMVB, a data-driven but simple probabilistic multi-variable branching method to accelerate mixed-integer programming. By constructing branching hyperplanes from probabilistic predictions and employing risk pooling tied to concentration inequalities, PMVB partitions the feasible region into subproblems that can be pruned efficiently or solved more rapidly. The framework supports both data-driven predictions and data-free LP-root-based surrogates, with theoretical justifications and practical demonstrations showing substantial speedups on synthetic benchmarks, real-world instances, and MIPLIB, across primal heuristics and branching roles. Its model-agnostic nature and minimal integration effort make PMVB a broadly applicable technique for speeding up online MIP solving in diverse domains.

Abstract

In this paper, we propose a Pre-trained Mixed Integer Optimization framework (PreMIO) that accelerates online mixed integer program (MIP) solving with offline datasets and machine learning models. Our method is based on a data-driven multi-variable cardinality branching procedure that splits the MIP feasible region using hyperplanes chosen by the concentration inequalities. Unlike most previous ML+MIP approaches that either require complicated implementation or suffer from a lack of theoretical justification, our method is simple, flexible, provable, and explainable. Numerical experiments on both classical OR benchmark datasets and real-life instances validate the efficiency of our proposed method.

Data-driven Mixed Integer Optimization through Probabilistic Multi-variable Branching

TL;DR

This work presents PMVB, a data-driven but simple probabilistic multi-variable branching method to accelerate mixed-integer programming. By constructing branching hyperplanes from probabilistic predictions and employing risk pooling tied to concentration inequalities, PMVB partitions the feasible region into subproblems that can be pruned efficiently or solved more rapidly. The framework supports both data-driven predictions and data-free LP-root-based surrogates, with theoretical justifications and practical demonstrations showing substantial speedups on synthetic benchmarks, real-world instances, and MIPLIB, across primal heuristics and branching roles. Its model-agnostic nature and minimal integration effort make PMVB a broadly applicable technique for speeding up online MIP solving in diverse domains.

Abstract

In this paper, we propose a Pre-trained Mixed Integer Optimization framework (PreMIO) that accelerates online mixed integer program (MIP) solving with offline datasets and machine learning models. Our method is based on a data-driven multi-variable cardinality branching procedure that splits the MIP feasible region using hyperplanes chosen by the concentration inequalities. Unlike most previous ML+MIP approaches that either require complicated implementation or suffer from a lack of theoretical justification, our method is simple, flexible, provable, and explainable. Numerical experiments on both classical OR benchmark datasets and real-life instances validate the efficiency of our proposed method.
Paper Structure (27 sections, 8 theorems, 43 equations, 4 figures, 4 tables, 1 algorithm)

This paper contains 27 sections, 8 theorems, 43 equations, 4 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Data-driven probabilistic multi-variable branching
  4. Parametric MIP and solution map
  5. Workflow of standard data-driven approaches for MIP.
  6. Statistical learning theory for solving MIP
  7. Risk-pooling and multi-variable cardinality branching
  8. Practical aspects of PMVB
  9. Practical probabilistic multi-variable cardinality branching
  10. Efficient pruning using objective cuts
  11. Compatibility with existing learning models
  12. Data-free PMVB using LP relaxation
  13. Analysis of data-free PMVB
  14. Using interior point solution in PMVB
  15. Numerical experiment
  16. ...and 12 more sections

Key Result

Lemma 2.1

Under A1 and A2, letting $\hat{y}_j$ be the ERM classifier under 0-1 loss and $e_j$ be the ERM error, then with probability at least $1 - \delta$, where $\operatorname{vc} (\mathcal{Y}_j)$ is the VC dimension of $\mathcal{Y}_j$.

Figures (4)

  • Figure 1: The mean and variance of $\alpha_{\mathcal{L}}(\tau; \xi)$ and $\alpha_{\mathcal{U}} (\tau; \xi)$, along with the average sizes of $\mathcal{L} (\tau; \xi)$ and $\mathcal{U} (\tau; \xi)$, computed over 100 independent set instances for varying values of the threshold parameter $\tau$.
  • Figure 2: Experiments using logistic regression model. From top to bottom: speedup on MKP, SCP and SCUC instances. Left: speedup of Gurobi, Right: speedup of COPT. Each tuple in the x-axis represents $(m, n, \text{Heuristic})$ and y-axis denotes average speedup $\Sigma$ in the corresponding settings.
  • Figure 3: Experiments using GNN on SCP, IS, and CA
  • Figure 4: Data-free experiments on SCP and CA

Theorems & Definitions (14)

  • Lemma 2.1: vapnik2013nature
  • Remark 2.1
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 3.1
  • Remark 3.1
  • Theorem 4.1
  • proof
  • Lemma A.1: Hoeffding inequality
  • ...and 4 more