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A generic Branch-and-Cut algorithm for bi-objective binary linear programs

Pierre Fouilhoux, Lucas Létocart, Yue Zhang

TL;DR

This paper presents the first generic bi-objective binary linear branch-and-cut algorithm to invoke state-of-the-art integer linear programming solver's internal advanced techniques such as cut separators.

Abstract

This paper presents the first generic bi-objective binary linear branch-and-cut algorithm. Studying the impact of valid inequalities in solution and objective spaces, two cutting frameworks are proposed. The multi-point separation problem is introduced together with a cutting algorithm to efficiently generate valid inequalities violating multiple points simultaneously. The other main idea is to invoke state-of-the-art integer linear programming solver's internal advanced techniques such as cut separators. Aggregation techniques are proposed to use these frameworks with a trade-off among efficient cut separations, tight lower and upper bound sets and advanced branching strategies. Experiments on various types of instances in the literature exhibit the promising efficiency of the algorithm that solves instances with up to 2800 binary variables in less than one hour of CPU time. Our algorithms are easy to extend for more than two objectives and integer variables.

A generic Branch-and-Cut algorithm for bi-objective binary linear programs

TL;DR

This paper presents the first generic bi-objective binary linear branch-and-cut algorithm to invoke state-of-the-art integer linear programming solver's internal advanced techniques such as cut separators.

Abstract

This paper presents the first generic bi-objective binary linear branch-and-cut algorithm. Studying the impact of valid inequalities in solution and objective spaces, two cutting frameworks are proposed. The multi-point separation problem is introduced together with a cutting algorithm to efficiently generate valid inequalities violating multiple points simultaneously. The other main idea is to invoke state-of-the-art integer linear programming solver's internal advanced techniques such as cut separators. Aggregation techniques are proposed to use these frameworks with a trade-off among efficient cut separations, tight lower and upper bound sets and advanced branching strategies. Experiments on various types of instances in the literature exhibit the promising efficiency of the algorithm that solves instances with up to 2800 binary variables in less than one hour of CPU time. Our algorithms are easy to extend for more than two objectives and integer variables.

Paper Structure

This paper contains 35 sections, 3 theorems, 1 equation, 5 figures, 7 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Bi-Objective Branch-and-Bound algorithm
  3. Ingredients of BOBLB&B algorithm
  4. Maintenance of bound sets
  5. Local relaxed bound set
  6. Global upper bound set
  7. Three rules of node fathoming
  8. Fathoming by infeasibility
  9. Fathoming by integrity
  10. Fathoming by dominance
  11. Branching
  12. Branching on variables
  13. Branching on criteria space (Pareto branching)
  14. Branching strategy
  15. Choice of the active node
  16. ...and 20 more sections

Key Result

Lemma 2.1

At the end of BOBLB&B algorithm, all integer equivalent solutions of $\mathcal{S}(\mathcal{U})$ is $\mathcal{X}_E$ the complete efficient set.

Figures (5)

  • Figure 1: New LBS of the intersection between $\widetilde{\mathcal{Y}}_N$ and $Dom(\mathcal{P})$
  • Figure 2: An example of the multi-point separation in bi-criteria space.
  • Figure 3: The new lower bound set invoking the ILP solver for cutting approach.
  • Figure 4: The two cases of $y^\lambda_{limit}$ induced by \ref{['algo:blackbox_LBS_cut']}.
  • Figure 5: The BOBLB&C with EPB and ISC performance evolution according to the number of $\Lambda$ parameters limited on each node for BOSCP instances .

Theorems & Definitions (6)

  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof