Journal of Global Optimization

, Volume 59, Issue 4, pp 811–836

# Benson type algorithms for linear vector optimization and applications

• Andreas H. Hamel
• Andreas Löhne
• Birgit Rudloff
Article

## Abstract

New versions and extensions of Benson’s outer approximation algorithm for solving linear vector optimization problems are presented. Primal and dual variants are provided in which only one scalar linear program has to be solved in each iteration rather than two or three as in previous versions. Extensions are given to problems with arbitrary pointed solid polyhedral ordering cones. Numerical examples are provided, one of them involving a new set-valued risk measure for multivariate positions.

## Keywords

Vector optimization Multiple objective optimization  Linear programming Duality Algorithms Outer approximation Set-valued risk measure Transaction costs

## Mathematics Subject Classification

90C29 90C05 90-08 91G99

## Notes

### Acknowledgments

We thank Dr Lizhen Shao for providing the data of Example 6.1 taken from [32] and we thank Professor Robert Vanderbei for supplying the data of Example 6.2 taken from [31].

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## Authors and Affiliations

• Andreas H. Hamel
• 1
• Andreas Löhne
• 2
Email author
• Birgit Rudloff
• 3
1. 1.Yeshiva UniversityNew YorkUSA
2. 2.Martin-Luther-Universität Halle-WittenbergGermany
3. 3.Princeton UniversityPrincetonUSA