IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v205y2025i1d10.1007_s10957-025-02626-3.html
   My bibliography  Save this article

A Method for Uncertain Linear Optimization Problems Through Polytopic Approximation of the Uncertainty Set

Author

Listed:
  • Ravi Raushan

    (Indian Institute of Technology (BHU))

  • Debdas Ghosh

    (Indian Institute of Technology (BHU))

  • Yong Zhao

    (Chongqing Jiaotong University)

  • Zhou Wei

    (Hebei University)

Abstract

In this work, we propose a globally convergent iterative method to solve uncertain constrained linear optimization problems. Due to the nondeterministic nature of such a problem, we use the min-max approach to convert the given problem into a deterministic one. We show that the robust feasible sets of the problem corresponding to the uncertainty set and the convex hull of the uncertainty set are identical. This result helps to reduce the number of inequality constraints of the problem drastically; often, this result reduces the semi-infinite programming problem of the min-max robust counterpart into a problem with a finite number of constraints. Following this, we provide a necessary and sufficient condition for the boundedness of the robust feasible set of the problem. Moreover, we explicitly identify the robust feasible set of the problem for polytopic and ellipsoidal uncertainty sets. We present an algorithm to construct an inner polytope of the convex hull of a general uncertainty set under a certain assumption. This algorithm provides a point-wise inner polytopic approximation of the convex hull with arbitrarily small precision. We employ this inner polytopic approximation corresponding to the uncertainty set and the infeasible interior-point technique to derive an iterative approach to solve general uncertain constrained linear optimization problems. Global convergence for the proposed method is reported. Numerical experiments illustrate the practical behaviour of the proposed method on discrete, star-shaped, disc-shaped, and ellipsoidal uncertainty sets.

Suggested Citation

  • Ravi Raushan & Debdas Ghosh & Yong Zhao & Zhou Wei, 2025. "A Method for Uncertain Linear Optimization Problems Through Polytopic Approximation of the Uncertainty Set," Journal of Optimization Theory and Applications, Springer, vol. 205(1), pages 1-42, April.
  • Handle: RePEc:spr:joptap:v:205:y:2025:i:1:d:10.1007_s10957-025-02626-3
    DOI: 10.1007/s10957-025-02626-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-025-02626-3
    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/s10957-025-02626-3?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

    for a different version of it.

    References listed on IDEAS

    as
    1. Dimitris Bertsimas & Omid Nohadani & Kwong Meng Teo, 2010. "Robust Optimization for Unconstrained Simulation-Based Problems," Operations Research, INFORMS, vol. 58(1), pages 161-178, February.
    2. Inuiguchi, Masahiro & Sakawa, Masatoshi, 1995. "Minimax regret solution to linear programming problems with an interval objective function," European Journal of Operational Research, Elsevier, vol. 86(3), pages 526-536, November.
    3. Huixian Wu & Hezhi Luo & Xianye Zhang & Haiqiang Qi, 2023. "An effective global algorithm for worst-case linear optimization under polyhedral uncertainty," Journal of Global Optimization, Springer, vol. 87(1), pages 191-219, September.
    4. 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.
    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. Zhou, Feng & Huang, Gordon H. & Chen, Guo-Xian & Guo, Huai-Cheng, 2009. "Enhanced-interval linear programming," European Journal of Operational Research, Elsevier, vol. 199(2), pages 323-333, December.
    2. Giove, Silvio & Funari, Stefania & Nardelli, Carla, 2006. "An interval portfolio selection problem based on regret function," European Journal of Operational Research, Elsevier, vol. 170(1), pages 253-264, April.
    3. J. Lasserre, 2011. "Min-max and robust polynomial optimization," Journal of Global Optimization, Springer, vol. 51(1), pages 1-10, September.
    4. Hong-Zhi Wei & Chun-Rong Chen & Sheng-Jie Li, 2020. "Robustness Characterizations for Uncertain Optimization Problems via Image Space Analysis," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 459-479, August.
    5. Wu, Hsien-Chung, 2009. "The Karush-Kuhn-Tucker optimality conditions in multiobjective programming problems with interval-valued objective functions," European Journal of Operational Research, Elsevier, vol. 196(1), pages 49-60, July.
    6. S. Rivaz & M. Yaghoobi, 2013. "Minimax regret solution to multiobjective linear programming problems with interval objective functions coefficients," 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. 21(3), pages 625-649, September.
    7. Engau, Alexander & Sigler, Devon, 2020. "Pareto solutions in multicriteria optimization under uncertainty," European Journal of Operational Research, Elsevier, vol. 281(2), pages 357-368.
    8. Kuntal Som & V. Vetrivel, 2023. "Global well-posedness of set-valued optimization with application to uncertain problems," Journal of Global Optimization, Springer, vol. 85(2), pages 511-539, February.
    9. Mausser, Helmut E. & Laguna, Manuel, 1999. "A heuristic to minimax absolute regret for linear programs with interval objective function coefficients," European Journal of Operational Research, Elsevier, vol. 117(1), pages 157-174, August.
    10. Javier León & Justo Puerto & Begoña Vitoriano, 2020. "A Risk-Aversion Approach for the Multiobjective Stochastic Programming Problem," Mathematics, MDPI, vol. 8(11), pages 1-26, November.
    11. S. Rivaz & M. A. Yaghoobi & M. Hladík, 2016. "Using modified maximum regret for finding a necessarily efficient solution in an interval MOLP problem," Fuzzy Optimization and Decision Making, Springer, vol. 15(3), pages 237-253, September.
    12. Tim Holzmann & J. Cole Smith, 2019. "Shortest path interdiction problem with arc improvement recourse: A multiobjective approach," Naval Research Logistics (NRL), John Wiley & Sons, vol. 66(3), pages 230-252, April.
    13. Lin, Jun & Ng, Tsan Sheng, 2011. "Robust multi-market newsvendor models with interval demand data," European Journal of Operational Research, Elsevier, vol. 212(2), pages 361-373, July.
    14. Botte, Marco & Schöbel, Anita, 2019. "Dominance for multi-objective robust optimization concepts," European Journal of Operational Research, Elsevier, vol. 273(2), pages 430-440.
    15. Chen Bai & Lixiao Yao & Cheng Wang & Yongxuan Zhao & Weien Peng, 2022. "Optimization of Water and Energy Spatial Patterns in the Cascade Pump Station Irrigation District," Sustainability, MDPI, vol. 14(9), pages 1-17, April.
    16. Gabrel, Virginie & Murat, Cécile & Thiele, Aurélie, 2014. "Recent advances in robust optimization: An overview," European Journal of Operational Research, Elsevier, vol. 235(3), pages 471-483.
    17. Anderson, Edward & Zachary, Stan, 2023. "Minimax decision rules for planning under uncertainty: Drawbacks and remedies," European Journal of Operational Research, Elsevier, vol. 311(2), pages 789-800.
    18. Oliveira, Carla & Antunes, Carlos Henggeler, 2007. "Multiple objective linear programming models with interval coefficients - an illustrated overview," European Journal of Operational Research, Elsevier, vol. 181(3), pages 1434-1463, September.
    19. Henriques, C.O. & Luque, M. & Marcenaro-Gutierrez, O.D. & Lopez-Agudo, L.A., 2019. "A multiobjective interval programming model to explore the trade-offs among different aspects of job satisfaction under different scenarios," Socio-Economic Planning Sciences, Elsevier, vol. 66(C), pages 35-46.
    20. Ying Cui & Ziyu He & Jong-Shi Pang, 2021. "Nonconvex robust programming via value-function optimization," Computational Optimization and Applications, Springer, vol. 78(2), pages 411-450, March.

    More about this item

    Keywords

    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    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:205:y:2025:i:1:d:10.1007_s10957-025-02626-3. 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.