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Computing the recession cone of a convex upper image via convex projection

Author

Listed:
  • Gabriela Kováčová

    (University of California)

  • Firdevs Ulus

    (Bilkent University)

Abstract

It is possible to solve unbounded convex vector optimization problems (CVOPs) in two phases: (1) computing or approximating the recession cone of the upper image and (2) solving the equivalent bounded CVOP where the ordering cone is extended based on the first phase. In this paper, we consider unbounded CVOPs and propose an alternative solution methodology to compute or approximate the recession cone of the upper image. In particular, we relate the dual of the recession cone with the Lagrange dual of weighted sum scalarization problems whenever the dual problem can be written explicitly. Computing this set requires solving a convex (or polyhedral) projection problem. We show that this methodology can be applied to semidefinite, quadratic, and linear vector optimization problems and provide some numerical examples.

Suggested Citation

  • Gabriela Kováčová & Firdevs Ulus, 2024. "Computing the recession cone of a convex upper image via convex projection," Journal of Global Optimization, Springer, vol. 89(4), pages 975-994, August.
    • Handle: RePEc:spr:jglopt:v:89:y:2024:i:4:d:10.1007_s10898-023-01351-3
    DOI: 10.1007/s10898-023-01351-3
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    References listed on IDEAS

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    1. Gabriela Kováčová & Birgit Rudloff, 2022. "Convex projection and convex multi-objective optimization," Journal of Global Optimization, Springer, vol. 83(2), pages 301-327, June.
    2. László Csirmaz, 2016. "Using multiobjective optimization to map the entropy region," Computational Optimization and Applications, Springer, vol. 63(1), pages 45-67, January.
    3. 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.
    4. T. D. Chuong & V. H. Mak-Hau & J. Yearwood & R. Dazeley & M.-T. Nguyen & T. Cao, 2022. "Robust Pareto solutions for convex quadratic multiobjective optimization problems under data uncertainty," Annals of Operations Research, Springer, vol. 319(2), pages 1533-1564, December.
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    6. Pubudu L. W. Jayasekara & Andrew C. Pangia & Margaret M. Wiecek, 2023. "On solving parametric multiobjective quadratic programs with parameters in general locations," Annals of Operations Research, Springer, vol. 320(1), pages 123-172, January.
    7. 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.
    8. 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.
    9. 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.
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    11. 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.
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