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Approximating biobjective minimization problems using general ordering cones

Author

Listed:
  • Arne Herzel

    (RPTU Kaiserslautern-Landau
    Weihenstephan-Triesdorf University of Applied Sciences)

  • Stephan Helfrich

    (RPTU Kaiserslautern-Landau)

  • Stefan Ruzika

    (RPTU Kaiserslautern-Landau)

  • Clemens Thielen

    (Weihenstephan-Triesdorf University of Applied Sciences
    Technical University of Munich)

Abstract

This article investigates the approximation quality achievable for biobjective minimization problems with respect to the Pareto cone by solutions that are (approximately) optimal with respect to larger ordering cones. When simultaneously considering $$\alpha $$ α -approximations for all closed convex ordering cones of a fixed inner angle $$\gamma \in \left[ \frac{\pi }{2}, \pi \right] $$ γ ∈ π 2 , π , an approximation guarantee between $$\alpha $$ α and $$2 \alpha $$ 2 α is achieved, which depends continuously on $$\gamma $$ γ . The analysis is best-possible for any inner angle and it generalizes and unifies the known results that the set of supported solutions is a 2-approximation and that the efficient set itself is a 1-approximation. Moreover, it is shown that, for maximization problems, no approximation guarantee is achievable in general by considering larger ordering cones in the described fashion, which again generalizes a known result about the set of supported solutions.

Suggested Citation

  • Arne Herzel & Stephan Helfrich & Stefan Ruzika & Clemens Thielen, 2023. "Approximating biobjective minimization problems using general ordering cones," Journal of Global Optimization, Springer, vol. 86(2), pages 393-415, June.
  • Handle: RePEc:spr:jglopt:v:86:y:2023:i:2:d:10.1007_s10898-023-01276-x
    DOI: 10.1007/s10898-023-01276-x
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    References listed on IDEAS

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    1. A. Engau & M. M. Wiecek, 2007. "Cone Characterizations of Approximate Solutions in Real Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 134(3), pages 499-513, September.
    2. Hunt, Brian J. & Wiecek, Margaret M. & Hughes, Colleen S., 2010. "Relative importance of criteria in multiobjective programming: A cone-based approach," European Journal of Operational Research, Elsevier, vol. 207(2), pages 936-945, December.
    3. Arne Herzel & Cristina Bazgan & Stefan Ruzika & Clemens Thielen & Daniel Vanderpooten, 2021. "One-exact approximate Pareto sets," Journal of Global Optimization, Springer, vol. 80(1), pages 87-115, May.
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