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Parallel Graver Basis Extraction for Nonlinear Integer Optimization

Wenbo Liu, Akang Wang, Wenguo Yang

TL;DR

A massively parallel heuristic for approximating Graver basis is developed, extracting promising directions by optimizing nonconvex continuous problems using parallelizable first-order methods.

Abstract

The augmentation scheme provides a nontraditional approach to nonlinear integer programming by iteratively refining incumbent solutions along objective-improving directions from the Graver basis. Its main computational bottleneck, however, lies in the practical difficulty of accessing such directions. To address this challenge, we develop a massively parallel heuristic for approximating Graver basis, extracting promising directions by optimizing nonconvex continuous problems using parallelizable first-order methods. Experiments on QPLIB and MINLPLib instances show that our method achieves comparable performance to advanced solvers.

Parallel Graver Basis Extraction for Nonlinear Integer Optimization

TL;DR

A massively parallel heuristic for approximating Graver basis is developed, extracting promising directions by optimizing nonconvex continuous problems using parallelizable first-order methods.

Abstract

The augmentation scheme provides a nontraditional approach to nonlinear integer programming by iteratively refining incumbent solutions along objective-improving directions from the Graver basis. Its main computational bottleneck, however, lies in the practical difficulty of accessing such directions. To address this challenge, we develop a massively parallel heuristic for approximating Graver basis, extracting promising directions by optimizing nonconvex continuous problems using parallelizable first-order methods. Experiments on QPLIB and MINLPLib instances show that our method achieves comparable performance to advanced solvers.

Paper Structure

This paper contains 17 sections, 4 theorems, 10 equations, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. The Graver Augmentation
  3. MAPE: Multi-start Augmentation via Parallel Extraction
  4. Characterization of $\mathrm{Ker}_\mathbb Z(A)$
  5. Parallel Extraction of the Graver Basis
  6. Multi-start Augmentation via Parallel Extraction
  7. Extension to Cases with Linear Inequalities
  8. Experiments
  9. Setup
  10. Datasets
  11. Baselines
  12. Evaluation Configuration
  13. Results and Analysis
  14. Metric
  15. Analysis
  16. ...and 2 more sections

Key Result

Proposition 2.1

The Graver basis $\mathcal{G}(A)$ forms a test set for model (eq:INLP) when the objective function $f: \mathbb{R}^n \to \mathbb{R}$ is separable convex, i.e., when $f(x) = \sum_{j=1}^n f_j(x_j)$ where each $f_j: \mathbb{R} \to \mathbb{R}$ is convex.

Theorems & Definitions (6)

  • Definition 2.1
  • Proposition 2.1: de2012algebraic
  • Lemma 3.1: kannan1979polynomial
  • Theorem 3.2
  • proof
  • Corollary 3.3