IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v167y2015i3d10.1007_s10957-013-0421-6.html
   My bibliography  Save this article

On Robust Optimization

Author

Listed:
  • Elisabeth Köbis

    (Martin-Luther-University Halle-Wittenberg)

Abstract

We introduce an unconstrained multicriteria optimization problem and discuss its relation to various well-known scalar robust optimization problems with a finite uncertainty set. Specifically, we show that a unique solution of a robust optimization problem is Pareto optimal for the unconstrained optimization problem. Furthermore, it is demonstrated that the set of weakly Pareto optimal solutions of the unconstrained multicriteria optimization problem contains all solutions of certain scalar robust optimization problems. An example is presented to verify our results. In addition, we show that the set of solutions of a weighted robust optimization problem always contains Pareto optimal solutions of the unconstrained multicriteria optimization problem. Similarly, we indicate that the set of solutions of a strictly robust optimization problem comprises Pareto optimal points of the unconstrained vector-valued problem. By assembling all these results we point out strong relations between unconstrained vector optimization and the more intuitively introduced concepts of scalar robust optimization. Finally, we provide a sufficient condition for an optimal solution of a strictly robust optimization problem.

Suggested Citation

  • Elisabeth Köbis, 2015. "On Robust Optimization," Journal of Optimization Theory and Applications, Springer, vol. 167(3), pages 969-984, December.
    • Handle: RePEc:spr:joptap:v:167:y:2015:i:3:d:10.1007_s10957-013-0421-6
    DOI: 10.1007/s10957-013-0421-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-013-0421-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-013-0421-6?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Serpil Say{i}n & Panos Kouvelis, 2005. "The Multiobjective Discrete Optimization Problem: A Weighted Min-Max Two-Stage Optimization Approach and a Bicriteria Algorithm," Management Science, INFORMS, vol. 51(10), pages 1572-1581, October.
    2. ,, 2000. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 16(2), pages 287-299, April.
    3. A. L. Soyster, 1973. "Technical Note—Convex Programming with Set-Inclusive Constraints and Applications to Inexact Linear Programming," Operations Research, INFORMS, vol. 21(5), pages 1154-1157, October.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Jonas Ide & Anita Schöbel, 2016. "Robustness for uncertain multi-objective optimization: a survey and analysis of different concepts," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 38(1), pages 235-271, January.
    2. Wenqing Chen & Melvyn Sim & Jie Sun & Chung-Piaw Teo, 2010. "From CVaR to Uncertainty Set: Implications in Joint Chance-Constrained Optimization," Operations Research, INFORMS, vol. 58(2), pages 470-485, April.
    3. Stefan Mišković, 2017. "A VNS-LP algorithm for the robust dynamic maximal covering location problem," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 39(4), pages 1011-1033, October.
    4. Antonio G. Martín & Manuel Díaz-Madroñero & Josefa Mula, 2020. "Master production schedule using robust optimization approaches in an automobile second-tier supplier," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 28(1), pages 143-166, March.
    5. Rashed Khanjani-Shiraz & Ali Babapour-Azar & Zohreh Hosseini-Noudeh & Panos M. Pardalos, 2022. "Distributionally robust maximum probability shortest path problem," Journal of Combinatorial Optimization, Springer, vol. 43(1), pages 140-167, January.
    6. Roberto Gomes de Mattos & Fabricio Oliveira & Adriana Leiras & Abdon Baptista de Paula Filho & Paulo Gonçalves, 2019. "Robust optimization of the insecticide-treated bed nets procurement and distribution planning under uncertainty for malaria prevention and control," Annals of Operations Research, Springer, vol. 283(1), pages 1045-1078, December.
    7. Fakhar, Majid & Mahyarinia, Mohammad Reza & Zafarani, Jafar, 2018. "On nonsmooth robust multiobjective optimization under generalized convexity with applications to portfolio optimization," European Journal of Operational Research, Elsevier, vol. 265(1), pages 39-48.
    8. Bastian, Nathaniel D. & Lunday, Brian J. & Fisher, Christopher B. & Hall, Andrew O., 2020. "Models and methods for workforce planning under uncertainty: Optimizing U.S. Army cyber branch readiness and manning," Omega, Elsevier, vol. 92(C).
    9. Klamroth, Kathrin & Köbis, Elisabeth & Schöbel, Anita & Tammer, Christiane, 2017. "A unified approach to uncertain optimization," European Journal of Operational Research, Elsevier, vol. 260(2), pages 403-420.
    10. Claire Nicolas & Stéphane Tchung-Ming & Emmanuel Hache, 2016. "Energy transition in transportation under cost uncertainty, an assessment based on robust optimization," Working Papers hal-02475943, HAL.
    11. Krumke, Sven O. & Schmidt, Eva & Streicher, Manuel, 2019. "Robust multicovers with budgeted uncertainty," European Journal of Operational Research, Elsevier, vol. 274(3), pages 845-857.
    12. Ghazaleh Ahmadi & Reza Tavakkoli-Moghaddam & Armand Baboli & Mehdi Najafi, 2022. "A decision support model for robust allocation and routing of search and rescue resources after earthquake: a case study," Operational Research, Springer, vol. 22(2), pages 1039-1081, April.
    13. Botte, Marco & Schöbel, Anita, 2019. "Dominance for multi-objective robust optimization concepts," European Journal of Operational Research, Elsevier, vol. 273(2), pages 430-440.
    14. Jiawei Chen & Suliman Al-Homidan & Qamrul Hasan Ansari & Jun Li & Yibing Lv, 2021. "Robust Necessary Optimality Conditions for Nondifferentiable Complex Fractional Programming with Uncertain Data," Journal of Optimization Theory and Applications, Springer, vol. 189(1), pages 221-243, April.
    15. Attila Bódis & János Balogh, 2019. "Bin packing problem with scenarios," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 27(2), pages 377-395, June.
    16. Heydari, Mohammadhossein & Sullivan, Kelly M., 2019. "Robust allocation of testing resources in reliability growth," Reliability Engineering and System Safety, Elsevier, vol. 192(C).
    17. Alan L. Erera & Juan C. Morales & Martin Savelsbergh, 2009. "Robust Optimization for Empty Repositioning Problems," Operations Research, INFORMS, vol. 57(2), pages 468-483, April.
    18. Oğuz Solyalı & Jean-François Cordeau & Gilbert Laporte, 2012. "Robust Inventory Routing Under Demand Uncertainty," Transportation Science, INFORMS, vol. 46(3), pages 327-340, August.
    19. Nicholas G. Hall & Daniel Zhuoyu Long & Jin Qi & Melvyn Sim, 2015. "Managing Underperformance Risk in Project Portfolio Selection," Operations Research, INFORMS, vol. 63(3), pages 660-675, June.
    20. Pätzold, Julius & Schöbel, Anita, 2020. "Approximate cutting plane approaches for exact solutions to robust optimization problems," European Journal of Operational Research, Elsevier, vol. 284(1), pages 20-30.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:167:y:2015:i:3:d:10.1007_s10957-013-0421-6. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.