Generator sets for the Minkowski sum problem
This paper develops a new theoretical framework for generator sets in the context of Minkowski Sum Problems (MSPs) arising in multi-objective optimization. In MSPs, the nondominated set of the global problem equals the nondominated set of the Minkowski sum of several local nondominated sets. We intr...
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| Vydané v: | European journal of operational research Ročník 328; číslo 3; s. 912 - 924 |
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| Jazyk: | English |
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Elsevier B.V
01.02.2026
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| ISSN: | 0377-2217 |
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| Abstract | This paper develops a new theoretical framework for generator sets in the context of Minkowski Sum Problems (MSPs) arising in multi-objective optimization. In MSPs, the nondominated set of the global problem equals the nondominated set of the Minkowski sum of several local nondominated sets. We introduce the concept of generator sets: subsets of local nondominated vectors that are sufficient to construct the global nondominated set. We present novel theoretical results that characterize conditions for vectors to belong to a generator set or to be redundant. Moreover, we develop algorithms for finding generator sets and identifying redundant local vectors. Finally, we conduct extensive numerical experiments to test the impact of varying characteristics of the instances on the resulting global nondominated set and the number of redundant vectors.
•Minkowski sum problems relate to additively-separable multi-objective problems.•Analysis of necessary local solutions for generating the nondominated sum.•Algorithms for finding minimum generator sets given known local sets.•Empirical study showing properties resulting in many redundant local solutions. |
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| AbstractList | This paper develops a new theoretical framework for generator sets in the context of Minkowski Sum Problems (MSPs) arising in multi-objective optimization. In MSPs, the nondominated set of the global problem equals the nondominated set of the Minkowski sum of several local nondominated sets. We introduce the concept of generator sets: subsets of local nondominated vectors that are sufficient to construct the global nondominated set. We present novel theoretical results that characterize conditions for vectors to belong to a generator set or to be redundant. Moreover, we develop algorithms for finding generator sets and identifying redundant local vectors. Finally, we conduct extensive numerical experiments to test the impact of varying characteristics of the instances on the resulting global nondominated set and the number of redundant vectors.
•Minkowski sum problems relate to additively-separable multi-objective problems.•Analysis of necessary local solutions for generating the nondominated sum.•Algorithms for finding minimum generator sets given known local sets.•Empirical study showing properties resulting in many redundant local solutions. |
| Author | Lyngesen, Mark Nielsen, Lars Relund Gadegaard, Sune Lauth |
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| Cites_doi | 10.1002/mcda.1598 10.1016/j.cor.2005.10.003 10.1016/j.cor.2023.106506 10.1137/130929564 10.1016/j.ejor.2019.07.027 10.1109/TEVC.2013.2281533 10.1007/s10479-013-1356-x 10.1111/itor.13191 10.1007/s10957-010-9786-y 10.4236/jamp.2015.31008 |
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| Keywords | Decomposition Minkowski sum Multiple criteria analysis Multiple objective programming |
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| Title | Generator sets for the Minkowski sum problem |
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