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Exact Objective Space Contraction for the Preprocessing of Multi-objective Integer Programs

Stephanie Riedmüller, Thorsten Koch

TL;DR

Multi-objective integer programs suffer from numerical instability and long runtimes when objective coefficients are large or span wide ranges. The authors propose an exact coefficient-transformation approach that contracts the objective space while preserving dominance, implemented via a cutting-plane ILP, and compare it to a heuristic scaling method. They establish theoretical results on when contractions are possible, develop a cutting-plane algorithm for the exact approach, and demonstrate via computational studies that contraction improves stability and increases the number of distinct non-dominated solutions found by the Defining Point Algorithm. The work provides a principled preprocessing tool for MOIP that helps stabilize computations and informs when exact contraction versus scaling should be used in practice.

Abstract

Solving integer optimization problems with large or widely ranged objective coefficients can lead to numerical instability and increased runtimes. When the problem also involves multiple objectives, the impact of the objective coefficients on runtimes and numerical issues multiplies. We address this issue by transforming the coefficients of linear objective functions into smaller integer coefficients. To the best of our knowledge, this problem has not been defined before. Next to a straightforward scaling heuristic, we introduce a novel exact transformation approach for the preprocessing of multi-objective binary problems. In this exact approach, the large or widely ranged integer objective coefficients are transformed into the minimal integer objective coefficients that preserve the dominance relation of the points in the objective space. The transformation problem is solved with an integer programming formulation with an exponential number of constraints. We present a cutting-plane algorithm that can efficiently handle the problem size. In a first computational study, we analyze how often and in which settings the transformation actually leads to smaller coefficients. In a second study, we evaluate how the exact transformation and a typical scaling heuristic, when used as preprocessing, affect the runtime and numerical stability of the Defining Point Algorithm.

Exact Objective Space Contraction for the Preprocessing of Multi-objective Integer Programs

TL;DR

Multi-objective integer programs suffer from numerical instability and long runtimes when objective coefficients are large or span wide ranges. The authors propose an exact coefficient-transformation approach that contracts the objective space while preserving dominance, implemented via a cutting-plane ILP, and compare it to a heuristic scaling method. They establish theoretical results on when contractions are possible, develop a cutting-plane algorithm for the exact approach, and demonstrate via computational studies that contraction improves stability and increases the number of distinct non-dominated solutions found by the Defining Point Algorithm. The work provides a principled preprocessing tool for MOIP that helps stabilize computations and informs when exact contraction versus scaling should be used in practice.

Abstract

Solving integer optimization problems with large or widely ranged objective coefficients can lead to numerical instability and increased runtimes. When the problem also involves multiple objectives, the impact of the objective coefficients on runtimes and numerical issues multiplies. We address this issue by transforming the coefficients of linear objective functions into smaller integer coefficients. To the best of our knowledge, this problem has not been defined before. Next to a straightforward scaling heuristic, we introduce a novel exact transformation approach for the preprocessing of multi-objective binary problems. In this exact approach, the large or widely ranged integer objective coefficients are transformed into the minimal integer objective coefficients that preserve the dominance relation of the points in the objective space. The transformation problem is solved with an integer programming formulation with an exponential number of constraints. We present a cutting-plane algorithm that can efficiently handle the problem size. In a first computational study, we analyze how often and in which settings the transformation actually leads to smaller coefficients. In a second study, we evaluate how the exact transformation and a typical scaling heuristic, when used as preprocessing, affect the runtime and numerical stability of the Defining Point Algorithm.

Paper Structure

This paper contains 17 sections, 5 theorems, 31 equations, 7 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Literature review
  3. Contracting large objective spaces
  4. Heuristic approach: Scaling and rounding objective coefficients
  5. Exact approach: Mapping objective coefficients in binary programs
  6. Full model
  7. Cutting-plane algorithm
  8. Existence of contractable objective functions
  9. Negative coefficients
  10. Computational study
  11. Contraction of individual objectives
  12. Contraction of the objective space
  13. Instability in DPA
  14. Instances and method
  15. Results
  16. ...and 2 more sections

Key Result

Theorem 1.8

For sets $X, Y, Z \subseteq \mathbb{R}^p$ and for a vector-valued linear objective function $f: X \rightarrow \mathbb{R}^p$, let $\varphi: Y \rightarrow Z$ be a monotone transformation with $f(X) \subseteq Y$. Then the sets of efficient solutions of $\min_{x \in X} f(x)\text{ and of } \min_{x \in X}

Figures (7)

  • Figure 1: Independent scaling of vectors in the coefficients matrix can lead to changes in the Pareto front. The feasible point set in the objective space is depicted. The non-dominated points are not filled.
  • Figure 2: Non-dominance is not preserved under the scale-and-round heuristic.
  • Figure 3: Contraction of an unnecessarily large objective space for a binary linear problem with two objectives. The filled area represents the hypervolume of the feasible set in the objective space, and the hatched area represents the contracted space.
  • Figure 4: Runtime and size deviation (contraction factor multiplied by 100) of the contraction of objective coefficients depicted as a box plot per number of coefficients.
  • Figure 5: Runtime vs deviation in coefficient size (contraction factor multiplied with 100) for a single objective contraction.
  • ...and 2 more figures

Theorems & Definitions (23)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 13 more