A Gr\"obner bases methodology for solving multiobjective polynomial integer programs

TL;DR

This paper introduces an algebraic approach using Gr"obner bases to solve multiobjective polynomial integer programs, addressing a less-explored area with theoretical analysis and computational experiments.

Contribution

It proposes a novel methodology transforming polynomial multiobjective problems into systems of equations solvable by Gr"obner bases, expanding solution techniques for complex polynomial integer programs.

Findings

Different transformations yield varied solution methodologies.

The proposed methods are analyzed theoretically and tested computationally.

Results demonstrate the effectiveness of Gr"obner bases in this context.

Abstract

Multiobjective discrete programming is a well-known family of optimization problems with a large spectrum of applications. The linear case has been tackled by many authors during the last years. However, the polynomial case has not been deeply studied due to its theoretical and computational difficulties. This paper presents an algebraic approach for solving these problems. We propose a methodology based on transforming the polynomial optimization problem in the problem of solving one or more systems of polynomial equations and we use certain Gr\"obner bases to solve these systems. Different transformations give different methodologies that are analyzed and compared from a theoretical point of view and by some computational experiments via the algorithms that they induce.