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Robust combinatorial optimization under budgeted–ellipsoidal uncertainty

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  • Jannis Kurtz

    (RWTH Aachen University)

Abstract

In the field of robust optimization, uncertain data are modeled by uncertainty sets which contain all relevant outcomes of the uncertain problem parameters. The complexity of the related robust problem depends strongly on the shape of the chosen set. Two popular classes of uncertainty are budgeted uncertainty and ellipsoidal uncertainty. In this paper, we introduce a new uncertainty class which is a combination of both. More precisely, we consider ellipsoidal uncertainty sets with the additional restriction that at most a certain number of ellipsoid axes can be used at the same time to describe a scenario. We define a discrete and a convex variant of the latter set and prove that in both cases the corresponding robust min–max problem is NP-hard for several combinatorial problems. Furthermore, we prove that for uncorrelated budgeted–ellipsoidal uncertainty in both cases the min–max problem can be solved in polynomial time for several combinatorial problems if the number of axes which can be used at the same time is fixed. We derive exact solution methods and formulations for the problem which we test on random instances of the knapsack problem and of the shortest path problem.

Suggested Citation

  • Jannis Kurtz, 2018. "Robust combinatorial optimization under budgeted–ellipsoidal uncertainty," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 6(4), pages 315-337, December.
  • Handle: RePEc:spr:eurjco:v:6:y:2018:i:4:d:10.1007_s13675-018-0097-7
    DOI: 10.1007/s13675-018-0097-7
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    References listed on IDEAS

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    1. A. Ben-Tal & A. Nemirovski, 1998. "Robust Convex Optimization," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 769-805, November.
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    3. Dimitris Bertsimas & Melvyn Sim, 2004. "The Price of Robustness," Operations Research, INFORMS, vol. 52(1), pages 35-53, February.
    4. Grani A. Hanasusanto & Daniel Kuhn & Wolfram Wiesemann, 2015. "K -Adaptability in Two-Stage Robust Binary Programming," Operations Research, INFORMS, vol. 63(4), pages 877-891, August.
    5. Aissi, Hassene & Bazgan, Cristina & Vanderpooten, Daniel, 2009. "Min-max and min-max regret versions of combinatorial optimization problems: A survey," European Journal of Operational Research, Elsevier, vol. 197(2), pages 427-438, September.
    6. David Pisinger, 1999. "Core Problems in Knapsack Algorithms," Operations Research, INFORMS, vol. 47(4), pages 570-575, August.
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    Cited by:

    1. Marin Bougeret & Jérémy Omer & Michael Poss, 2023. "Optimization Problems in Graphs with Locational Uncertainty," INFORMS Journal on Computing, INFORMS, vol. 35(3), pages 578-592, May.
    2. Arie M. C. A. Koster & Michael Poss, 2018. "Special issue on: robust combinatorial optimization," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 6(3), pages 207-209, September.
    3. Christoph Buchheim & Dorothee Henke, 2022. "The robust bilevel continuous knapsack problem with uncertain coefficients in the follower’s objective," Journal of Global Optimization, Springer, vol. 83(4), pages 803-824, August.

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