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Adaptive Improvements of Multi-Objective Branch and Bound

Julius Bauß, Sophie N. Parragh, Michael Stiglmayr

TL;DR

The paper tackles the challenge of weakening bounds in multi-objective branch-and-bound as the number of objectives grows, by embedding objective-space information into decision-space search. It introduces a dynamic node selection strategy based on an approximate hypervolume gap and augments the bound sets by solving scalarizations to integer optimality, including warmstarting the bounds and using ε-constraint scalarization, to obtain both improved upper and lower bounds. Numerical experiments across knapsack, uncapacitated/capacitated facility location, and generalized assignment problems show substantial reductions in explored nodes and, in many cases, runtime, with performance depending on problem class and dimensionality. The work highlights that a carefully chosen mix of dynamic node selection, objective-space guidance, and selective scalarizations can meaningfully improve MO-BnB performance, while suggesting future exploration of hybrid, problem-dependent strategies.

Abstract

Branch and bound methods which are based on the principle "divide and conquer" are a well established solution approach in single-objective integer programming. In multi-objective optimization branch and bound algorithms are increasingly attracting interest. However, the larger number of objectives raises additional difficulties for implicit enumeration approaches like branch and bound. Since bounding and pruning is considerably weaker in multiple objectives, many branches have to be (partially) searched and may not be pruned directly. The adaptive use of objective space information can guide the search in promising directions to determine a good approximation of the Pareto front already in early stages of the algorithm. In particular we focus in this article on improving the branching and queuing of subproblems and the handling of lower bound sets. In our numerical test we evaluate the impact of the proposed methods in comparison to a standard implementation of multiobjective branch and bound on knapsack problems, generalized assignment problems and (un)capacitated facility location problems.

Adaptive Improvements of Multi-Objective Branch and Bound

TL;DR

The paper tackles the challenge of weakening bounds in multi-objective branch-and-bound as the number of objectives grows, by embedding objective-space information into decision-space search. It introduces a dynamic node selection strategy based on an approximate hypervolume gap and augments the bound sets by solving scalarizations to integer optimality, including warmstarting the bounds and using ε-constraint scalarization, to obtain both improved upper and lower bounds. Numerical experiments across knapsack, uncapacitated/capacitated facility location, and generalized assignment problems show substantial reductions in explored nodes and, in many cases, runtime, with performance depending on problem class and dimensionality. The work highlights that a carefully chosen mix of dynamic node selection, objective-space guidance, and selective scalarizations can meaningfully improve MO-BnB performance, while suggesting future exploration of hybrid, problem-dependent strategies.

Abstract

Branch and bound methods which are based on the principle "divide and conquer" are a well established solution approach in single-objective integer programming. In multi-objective optimization branch and bound algorithms are increasingly attracting interest. However, the larger number of objectives raises additional difficulties for implicit enumeration approaches like branch and bound. Since bounding and pruning is considerably weaker in multiple objectives, many branches have to be (partially) searched and may not be pruned directly. The adaptive use of objective space information can guide the search in promising directions to determine a good approximation of the Pareto front already in early stages of the algorithm. In particular we focus in this article on improving the branching and queuing of subproblems and the handling of lower bound sets. In our numerical test we evaluate the impact of the proposed methods in comparison to a standard implementation of multiobjective branch and bound on knapsack problems, generalized assignment problems and (un)capacitated facility location problems.
Paper Structure (24 sections, 10 equations, 2 figures, 4 tables, 1 algorithm)

This paper contains 24 sections, 10 equations, 2 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work: Multi-objective Branch and Bound Frameworks in the Literature
  3. Lower bound:
  4. Upper bound:
  5. Node selection:
  6. Fathoming:
  7. Branching:
  8. Using Objective Space Information in Multi-objective Branch and Bound
  9. A Dynamic Node Selection Strategy
  10. Improving Bounds by Solving Scalarizations to Integer Optimality
  11. Warmstarting the Bound Sets
  12. Improving the Upper Bound set by Using $\varepsilon$-constraint Scalarization
  13. Using Simple Lower Bound Sets
  14. Algorithmic Control of the Presented Approaches
  15. Numerical Tests
  16. ...and 9 more sections

Figures (2)

  • Figure 1: A bi-objective example of computation of the two different approximated hypervolume gap approaches. In (a) and (b), the the approximated hypervolume gap (gray) is visualized for the local upper bounds $lu^1$ and $lu^2$. In (c) and (d), the hypervolume of the box (gray) defined by the local ideal point and the local upper bound $lu^1$ respectively $lu^2$ is shown.
  • Figure 2: Performance profiles of different problem classes.