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Parametric Disjunctive Cuts for Sequences of Mixed Integer Linear Optimization Problems

Shannon Kelley, Aleksandr M. Kazachkov, Ted Ralphs

TL;DR

The paper addresses accelerating sequences of related MILPs by reusing disjunctive proofs through parametric disjunctive inequalities (PDIs). PDIs are constructed with parametric certificates derived from Farkas multipliers associated with large disjunctive terms, enabling valid cuts for perturbed instances without solving new cut-generating LPs. It provides sufficient conditions for hull support and a tightening step to recover it when needed, and integrates PDIs into a branch-and-cut framework. Empirical evaluation on perturbed MIPLIB 2017 instances shows that PDIs typically improve total solve time and often outperform non-parametric disjunctive cuts and the default solver configuration, with complementary strengths across perturbation types.

Abstract

Many applications require solving sequences of related mixed-integer linear programs. We introduce a class of parametric disjunctive inequalities (PDIs), obtained by reusing the disjunctive proofs of optimality from prior solves to construct cuts valid for perturbed instances. We describe several methods of generating such cuts that navigate the tradeoff between computational expense and strength. We provide sufficient conditions under which PDIs support the disjunctive hull and a tightening step that guarantees support when needed. On perturbed instances from MIPLIB 2017, augmenting branch-and-cut with PDIs substantially improves performance, reducing total solve times on the majority of challenging cases.

Parametric Disjunctive Cuts for Sequences of Mixed Integer Linear Optimization Problems

TL;DR

The paper addresses accelerating sequences of related MILPs by reusing disjunctive proofs through parametric disjunctive inequalities (PDIs). PDIs are constructed with parametric certificates derived from Farkas multipliers associated with large disjunctive terms, enabling valid cuts for perturbed instances without solving new cut-generating LPs. It provides sufficient conditions for hull support and a tightening step to recover it when needed, and integrates PDIs into a branch-and-cut framework. Empirical evaluation on perturbed MIPLIB 2017 instances shows that PDIs typically improve total solve time and often outperform non-parametric disjunctive cuts and the default solver configuration, with complementary strengths across perturbation types.

Abstract

Many applications require solving sequences of related mixed-integer linear programs. We introduce a class of parametric disjunctive inequalities (PDIs), obtained by reusing the disjunctive proofs of optimality from prior solves to construct cuts valid for perturbed instances. We describe several methods of generating such cuts that navigate the tradeoff between computational expense and strength. We provide sufficient conditions under which PDIs support the disjunctive hull and a tightening step that guarantees support when needed. On perturbed instances from MIPLIB 2017, augmenting branch-and-cut with PDIs substantially improves performance, reducing total solve times on the majority of challenging cases.

Paper Structure

This paper contains 33 sections, 5 theorems, 6 equations, 5 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Notation and Preliminaries
  4. Disjunctive Cut Generation.
  5. Parametric Disjunctive Cuts
  6. Certificates.
  7. Parametric Cut Generation.
  8. Validity.
  9. Strengthening.
  10. Step 1: Reparameterize to Find Supporting Certificates.
  11. Step 2: Regenerate to Yield Supporting PDIs.
  12. Computational Experiments
  13. Test Set Generation.
  14. Methods Compared.
  15. Software and Hardware Dependencies.
  16. ...and 18 more sections

Key Result

theorem 1

Let $k, \ell \in \mathcal{K}$. Then given a Farkas certificate $\{v^{t}\}_{t \in T}$ induced by disjunction $\{\mathcal{X}^{t}\}_{t \in T}$ and IP-$k$, we have that $\alpha^\top x \ge \beta$ for all $x \in \mathcal{P}_{D}^{\ell}$, where $\alpha_j \mathrel{\vcenter{\hbox{\footnotesize.}\hbox{\footnot

Figures (5)

  • Figure 1: Examples showing that PDIs from the same certificate need not all support the disjunctive hull
  • Figure 2: For matrix perturbations, PDIs often outperform VPCs (left), and the best disjunctive cut generator almost always beats default Gurobi (right)
  • Figure 3: PDIs often outperform VPCs (left) and default Gurobi (right) for objective perturbations
  • Figure 4: PDIs often outperform VPCs and default Gurobi for right-hand side perturbations
  • Figure 5: For perturbation-induced feasible terms

Theorems & Definitions (19)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • definition 5
  • definition 6
  • definition 7
  • definition 8
  • theorem 1
  • definition 9
  • ...and 9 more