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$\mathcal{V}$-Polyhedral Disjunctive Cuts

Egon Balas, Aleksandr M. Kazachkov

TL;DR

In the results, VPCs from a strong disjunction significantly improve the gap closed compared to existing cuts in solvers, and they also decrease some instances' solving time when used with branch and bound.

Abstract

We introduce $\mathcal{V}$-polyhedral disjunctive cuts (VPCs) for generating valid inequalities from general disjunctions. Cuts are critical to integer programming solvers, but the benefit from many families is only realized when the cuts are applied recursively, causing numerical instability and "tailing off" of cut strength after several rounds. To mitigate these difficulties, the VPC framework offers a practical method for generating strong cuts without resorting to recursion. The framework starts with a disjunction whose terms partition the feasible region into smaller subproblems, then obtains a collection of points and rays from the disjunctive terms, from which we build a linear program whose feasible solutions correspond to valid disjunctive cuts. Though a naïve implementation would result in an exponentially-sized optimization problem, we show how to efficiently construct this linear program, such that it is much smaller than the one from the alternative higher-dimensional cut-generating linear program. This enables us to test strong multiterm disjunctions that arise from the leaf nodes of a partial branch-and-bound tree. In addition to proving useful theoretical properties of the cuts, we evaluate their performance computationally through an implementation in the open-source COIN-OR framework. In the results, VPCs from a strong disjunction significantly improve the gap closed compared to existing cuts in solvers, and they also decrease some instances' solving time when used with branch and bound.

$\mathcal{V}$-Polyhedral Disjunctive Cuts

TL;DR

In the results, VPCs from a strong disjunction significantly improve the gap closed compared to existing cuts in solvers, and they also decrease some instances' solving time when used with branch and bound.

Abstract

We introduce -polyhedral disjunctive cuts (VPCs) for generating valid inequalities from general disjunctions. Cuts are critical to integer programming solvers, but the benefit from many families is only realized when the cuts are applied recursively, causing numerical instability and "tailing off" of cut strength after several rounds. To mitigate these difficulties, the VPC framework offers a practical method for generating strong cuts without resorting to recursion. The framework starts with a disjunction whose terms partition the feasible region into smaller subproblems, then obtains a collection of points and rays from the disjunctive terms, from which we build a linear program whose feasible solutions correspond to valid disjunctive cuts. Though a naïve implementation would result in an exponentially-sized optimization problem, we show how to efficiently construct this linear program, such that it is much smaller than the one from the alternative higher-dimensional cut-generating linear program. This enables us to test strong multiterm disjunctions that arise from the leaf nodes of a partial branch-and-bound tree. In addition to proving useful theoretical properties of the cuts, we evaluate their performance computationally through an implementation in the open-source COIN-OR framework. In the results, VPCs from a strong disjunction significantly improve the gap closed compared to existing cuts in solvers, and they also decrease some instances' solving time when used with branch and bound.
Paper Structure (38 sections, 11 theorems, 14 equations, 7 figures, 10 tables, 1 algorithm)

This paper contains 38 sections, 11 theorems, 14 equations, 7 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Paper organization.
  3. Notation.
  4. Point-ray linear program
  5. Normalization of the PRLP
  6. Proper point-ray collections: ensuring valid inequalities
  7. Simple VPCs from simple cone relaxations
  8. Simple VPCs corresponding to facets of the disjunctive hull
  9. Choosing strong disjunctions
  10. Choosing appropriate objectives
  11. Computational results
  12. Computational setup
  13. Instance selection.
  14. Generation of disjunction.
  15. Evaluation within branch and bound.
  16. ...and 23 more sections

Key Result

Theorem 1

The inequality $\alpha^\mathsf{ T} x \ge \beta$ is valid for $\mathop{\mathrm{conv}}\limits(\mathcal{P}) + \mathop{\mathrm{cone}}\limits(\mathcal{R})$ if and only if $(\alpha,\beta)$ is a feasible solution to PRLP. The inequality defines a facet of the point-ray hull if and only if the solution $(\a

Figures (7)

  • Figure 1: This example shows how the simple point-ray collection could limit the set of obtainable inequalities valid for the disjunctive hull. The right panel shows that the dashed inequality (that is valid for $P_I$) would violate a ray of the cone $C^2$. The cones are shown as two dimensional for ease of illustration.
  • Figure 2: Rays in the point-ray collection impact the set of cuts that can be generated. The dashed wavy line in the third panel corresponds to a ray $r$ that is added to $\mathcal{R}^0$ from one term of the disjunction, but affects the point-ray hull when it originates from a point from a different term.
  • Figure 3: The disjunction is all possible assignments of $x_1$ and $x_2$.
  • Figure 4: A different variable is branched on, resulting in some pruned nodes and two stronger disjunctive terms.
  • Figure 6: Example that shows an inequality tight at ${\ref{['p*']}}$ that does not cut away $\bar{x}$ may be necessary for achieving the bound $c^\mathsf{ T} {\ref{['p*']}}$. In this example, we assume we are maximizing along the vertical axis. The first panel shows the original polytope. The second panel is the polytope after adding the only split inequality that cuts away $\bar{x}$. The third panel shows the polytope after adding all the valid split cuts.
  • ...and 2 more figures

Theorems & Definitions (21)

  • Theorem 1
  • proof
  • Definition 2
  • Corollary 3
  • proof
  • Corollary 4
  • Corollary 5
  • Proposition 6
  • proof
  • Theorem 7
  • ...and 11 more