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

TL;DR

This paper introduces an exact transformation method to reduce large objective coefficients in multi-objective integer programs, improving numerical stability and runtime efficiency, and compares it with heuristic scaling approaches.

Contribution

The paper presents a novel exact transformation approach for preprocessing multi-objective binary problems, preserving dominance relations while minimizing objective coefficients.

Findings

The exact transformation often results in smaller coefficients.

Using the transformation improves numerical stability.

It reduces runtime in the Defining Point Algorithm.

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.