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A vector linear programming approach for certain global optimization problems

Author

Listed:
  • Daniel Ciripoi

    (Friedrich Schiller University Jena)

  • Andreas Löhne

    (Friedrich Schiller University Jena)

  • Benjamin Weißing

    (Friedrich Schiller University Jena)

Abstract

Global optimization problems with a quasi-concave objective function and linear constraints are studied. We point out that various other classes of global optimization problems can be expressed in this way. We present two algorithms, which can be seen as slight modifications of Benson-type algorithms for multiple objective linear programs (MOLP). The modification of the MOLP algorithms results in a more efficient treatment of the studied optimization problems. This paper generalizes results of Schulz and Mittal (Math Program 141(1–2):103–120, 2013) on quasi-concave problems and Shao and Ehrgott (Optimization 65(2):415–431, 2016) on multiplicative linear programs. Furthermore, it improves results of Löhne and Wagner (J Glob Optim 69(2):369–385, 2017) on minimizing the difference $$f=g-h$$ f = g - h of two convex functions g, h where either g or h is polyhedral. Numerical examples are given and the results are compared with the global optimization software BARON.

Suggested Citation

  • Daniel Ciripoi & Andreas Löhne & Benjamin Weißing, 2018. "A vector linear programming approach for certain global optimization problems," Journal of Global Optimization, Springer, vol. 72(2), pages 347-372, October.
    • Handle: RePEc:spr:jglopt:v:72:y:2018:i:2:d:10.1007_s10898-018-0627-0
    DOI: 10.1007/s10898-018-0627-0
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    References listed on IDEAS

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    1. Andreas Löhne & Andrea Wagner, 2017. "Solving DC programs with a polyhedral component utilizing a multiple objective linear programming solver," Journal of Global Optimization, Springer, vol. 69(2), pages 369-385, October.
    2. Andreas Hamel & Andreas Löhne & Birgit Rudloff, 2014. "Benson type algorithms for linear vector optimization and applications," Journal of Global Optimization, Springer, vol. 59(4), pages 811-836, August.
    3. Albert Ferrer & Adil Bagirov & Gleb Beliakov, 2015. "Solving DC programs using the cutting angle method," Journal of Global Optimization, Springer, vol. 61(1), pages 71-89, January.
    4. Matthias Ehrgott & Andreas Löhne & Lizhen Shao, 2012. "A dual variant of Benson’s “outer approximation algorithm” for multiple objective linear programming," Journal of Global Optimization, Springer, vol. 52(4), pages 757-778, April.
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    Cited by:

    1. Gabriela Kov'av{c}ov'a & Birgit Rudloff, 2018. "Time consistency of the mean-risk problem," Papers 1806.10981, arXiv.org, revised Jan 2020.
    2. Simeon vom Dahl & Andreas Löhne, 2020. "Solving polyhedral d.c. optimization problems via concave minimization," Journal of Global Optimization, Springer, vol. 78(1), pages 37-47, September.
    3. 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.

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