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Primal and dual approximation algorithms for convex vector optimization problems

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  • Andreas Löhne
  • Birgit Rudloff
  • Firdevs Ulus

Abstract

Two approximation algorithms for solving convex vector optimization problems (CVOPs) are provided. Both algorithms solve the CVOP and its geometric dual problem simultaneously. The first algorithm is an extension of Benson’s outer approximation algorithm, and the second one is a dual variant of it. Both algorithms provide an inner as well as an outer approximation of the (upper and lower) images. Only one scalar convex program has to be solved in each iteration. We allow objective and constraint functions that are not necessarily differentiable, allow solid pointed polyhedral ordering cones, and relate the approximations to an appropriate $$\epsilon $$ ϵ -solution concept. Numerical examples are provided. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Andreas Löhne & Birgit Rudloff & Firdevs Ulus, 2014. "Primal and dual approximation algorithms for convex vector optimization problems," Journal of Global Optimization, Springer, vol. 60(4), pages 713-736, December.
    • Handle: RePEc:spr:jglopt:v:60:y:2014:i:4:p:713-736
    DOI: 10.1007/s10898-013-0136-0
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    1. Lizhen Shao & Matthias Ehrgott, 2008. "Approximating the nondominated set of an MOLP by approximately solving its dual problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(3), pages 469-492, December.
    2. Lizhen Shao & Matthias Ehrgott, 2008. "Approximately solving multiobjective linear programmes in objective space and an application in radiotherapy treatment planning," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(2), pages 257-276, October.
    3. JosÉ Figueira & Salvatore Greco & Matthias Ehrogott, 2005. "Multiple Criteria Decision Analysis: State of the Art Surveys," International Series in Operations Research and Management Science, Springer, number 978-0-387-23081-8, March.
    4. Andreas H. Hamel & Birgit Rudloff & Mihaela Yankova, 2012. "Set-valued average value at risk and its computation," Papers 1202.5702, arXiv.org, revised Jan 2013.
    5. Matthias Ehrgott & Lizhen Shao & Anita Schöbel, 2011. "An approximation algorithm for convex multi-objective programming problems," Journal of Global Optimization, Springer, vol. 50(3), pages 397-416, July.
    6. Y.M. Kabanov, 1999. "Hedging and liquidation under transaction costs in currency markets," Finance and Stochastics, Springer, vol. 3(2), pages 237-248.
    7. S. Ruzika & M. M. Wiecek, 2005. "Approximation Methods in Multiobjective Programming," Journal of Optimization Theory and Applications, Springer, vol. 126(3), pages 473-501, September.
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    4. Zachary Feinstein & Birgit Rudloff, 2024. "Deep learning the efficient frontier of convex vector optimization problems," Journal of Global Optimization, Springer, vol. 90(2), pages 429-458, October.
    5. Daniel Dörfler, 2022. "On the Approximation of Unbounded Convex Sets by Polyhedra," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 265-287, July.
    6. Hadjer Belkhiri & Mohamed El-Amine Chergui & Fatma Zohra Ouaïl, 2022. "Optimizing a linear function over an efficient set," Operational Research, Springer, vol. 22(4), pages 3183-3201, September.
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    8. Birgit Rudloff & Firdevs Ulus, 2019. "Certainty Equivalent and Utility Indifference Pricing for Incomplete Preferences via Convex Vector Optimization," Papers 1904.09456, arXiv.org, revised Oct 2020.
    9. Robert Bassett & Khoa Le, 2016. "Multistage Portfolio Optimization: A Duality Result in Conic Market Models," Papers 1601.00712, arXiv.org, revised Jan 2016.
    10. Soghra Nobakhtian & Narjes Shafiei, 2017. "A Benson type algorithm for nonconvex multiobjective programming problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(2), pages 271-287, July.
    11. Ina Lammel & Karl-Heinz Küfer & Philipp Süss, 2024. "An approximation algorithm for multiobjective mixed-integer convex optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(1), pages 321-350, August.
    12. c{C}au{g}{i}n Ararat & Nurtai Meimanjan, 2019. "Computation of systemic risk measures: a mixed-integer programming approach," Papers 1903.08367, arXiv.org, revised Aug 2023.
    13. Gabriele Eichfelder & Leo Warnow, 2022. "An approximation algorithm for multi-objective optimization problems using a box-coverage," Journal of Global Optimization, Springer, vol. 83(2), pages 329-357, June.
    14. Zachary Feinstein & Birgit Rudloff, 2022. "Deep Learning the Efficient Frontier of Convex Vector Optimization Problems," Papers 2205.07077, arXiv.org, revised May 2024.
    15. Firdevs Ulus, 2018. "Tractability of convex vector optimization problems in the sense of polyhedral approximations," Journal of Global Optimization, Springer, vol. 72(4), pages 731-742, December.
    16. Gabriele Eichfelder & Oliver Stein & Leo Warnow, 2024. "A Solver for Multiobjective Mixed-Integer Convex and Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1736-1766, November.
    17. Zachary Feinstein & Birgit Rudloff, 2017. "A recursive algorithm for multivariate risk measures and a set-valued Bellman’s principle," Journal of Global Optimization, Springer, vol. 68(1), pages 47-69, May.
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