Generator Sets for the Minkowski Sum Problem -- Theory and Insights

TL;DR

This paper explores the structure of Minkowski sum problems in multi-objective optimization, analyzing redundant vectors and proposing an algorithm to improve computational efficiency through bounding sets.

Contribution

It introduces a theoretical analysis of necessary and redundant vectors in Minkowski sum problems and proposes an algorithm to identify and eliminate unnecessary local vectors.

Findings

Redundant vectors can be identified using bounding set techniques.

The proposed algorithm reduces computational efforts in generating the global nondominated set.

Numerical experiments show the impact of instance characteristics on redundancy and solution quality.

Abstract

This paper considers a class of multi-objective optimization problems known as Minkowski sum problems. Minkowski sum problems have a decomposable structure, where the global nondominated (Pareto) set corresponds to the Minkowski sum of several local nondominated sets. In some cases, the vectors of local sets does not contribute to the generation of the global nondominated set, and may therefore lead to wasted computational efforts. Therefore, we investigate theoretical properties of both necessary and redundant vectors, and propose an algorithm based on bounding sets for identifying unnecessary local vectors. We conduct extensive numerical experiments to test the the impact of varying characteristics of the instances on the resulting global nondominated set and the number of redundant vectors.