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Efficient Branching Rules for Optimizing Range and Order-Based Objective Functions

Bart van Rossum, Rui Chen, Andrea Lodi

TL;DR

Range branching is proposed, a generic branching rule that directly tackles range minimization problems featuring exponentially many variables and can be used on top of problem-specific branching schemes, and how it can be successfully generalized to order-based objective functions, such as the Gini deviation.

Abstract

We consider range minimization problems featuring exponentially many variables, as frequently arising in fairness-oriented or bi-objective optimization. While branch and price is successful at solving cost-oriented problems with many variables, the performance of classical branch-and-price algorithms for range minimization is drastically impaired by weak linear programming relaxations. We propose range branching, a generic branching rule that directly tackles this issue and can be used on top of problem-specific branching schemes. We show several desirable properties of range branching and show its effectiveness on a series of instances of the fair capacitated vehicle routing problem and fair generalized assignment problem. Range branching significantly improves multiple classical branching schemes in terms of computing time, optimality gap, and size of the branch-and-bound tree, allowing us to solve many more large instances than classical methods. Moreover, we show how range branching can be successfully generalized to order-based objective functions, such as the Gini deviation.

Efficient Branching Rules for Optimizing Range and Order-Based Objective Functions

TL;DR

Range branching is proposed, a generic branching rule that directly tackles range minimization problems featuring exponentially many variables and can be used on top of problem-specific branching schemes, and how it can be successfully generalized to order-based objective functions, such as the Gini deviation.

Abstract

We consider range minimization problems featuring exponentially many variables, as frequently arising in fairness-oriented or bi-objective optimization. While branch and price is successful at solving cost-oriented problems with many variables, the performance of classical branch-and-price algorithms for range minimization is drastically impaired by weak linear programming relaxations. We propose range branching, a generic branching rule that directly tackles this issue and can be used on top of problem-specific branching schemes. We show several desirable properties of range branching and show its effectiveness on a series of instances of the fair capacitated vehicle routing problem and fair generalized assignment problem. Range branching significantly improves multiple classical branching schemes in terms of computing time, optimality gap, and size of the branch-and-bound tree, allowing us to solve many more large instances than classical methods. Moreover, we show how range branching can be successfully generalized to order-based objective functions, such as the Gini deviation.
Paper Structure (39 sections, 6 theorems, 15 equations, 5 figures, 5 tables)

This paper contains 39 sections, 6 theorems, 15 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Range Minimization Problem
  3. Problem Description
  4. Mathematical Formulation
  5. Range Branching
  6. Range Branching Rule
  7. Properties of Range Branching
  8. Practical Considerations
  9. Cutoff Values
  10. Further Improvements
  11. Fair Capacitated Vehicle Routing Problem
  12. Mathematical Formulation
  13. Vehicle-Index Formulation
  14. Last-Customer Formulation
  15. Branch and Price
  16. ...and 24 more sections

Key Result

Proposition 2

Let $\mathcal{X}$ be such that there exist distinct $\bm{x}_1, \bm{x}_2 \in \mathcal{X}$ for which $p_{\text{max}}(\bm{x}_1) > p_{\text{max}}(\bm{x}_2)$ or $p_{\text{min}}(\bm{x}_1) > p_{\text{min}}(\bm{x}_2)$. Then, for any valid $\bm{w}$, it holds that $\mathcal{Z}_{LP}(\bm{w})$ contains infinitel

Figures (5)

  • Figure 1: Efficient (left) and fair (right) vehicle routing solutions.
  • Figure 2: Feasible region in the space of $(x_1, x_2)$ (left) and $(\eta, \gamma)$ (right). Integer-feasible points are indicated by black dots, and the feasible region of the LP relaxation is indicated in grey.
  • Figure 3: Example of range branching applied to a range-violating solution. The striped rectangle indicates columns, whose payoffs are above the threshold, that have been fixed to zero.
  • Figure 4: Progression of lower and upper bound of vehicle-index formulation (left) and last-customer formulation (right) with classical and range branching scheme.
  • Figure 5: Progression of lower and upper bound of vehicle-index formulation for Gini deviation with the classical and order branching scheme.

Theorems & Definitions (16)

  • Definition 1: Range-Respecting Solution
  • Definition 2: Valid Formulation
  • Proposition 2: Fractional Range-Violating Solutions
  • proof
  • Example 1
  • Proposition 3: RBF Node Solutions are Range-Respecting
  • proof
  • Corollary 4: Formulation-Independence
  • proof
  • Proposition 5
  • ...and 6 more